Vector

A Vector may be described as a quantity having both magnitude & direction. A vector is generally represented by a directed line segment, say (overrightarrow {AB}).  A is called the initial point & B is called the terminal point. The magnitude of vector (overrightarrow {AB}) is expressed b (|overrightarrow {AB}|)

Zero Vector  a vector of zero magnitude i.e.which has the same initial & terminal point, is called a Zero Vector. It is denoted by O. Unit Vector a vector of unit magnitude in direction of a vector (overrightarrow{mathrm{a}}) is called unit vector along (overrightarrow{mathrm{a}}) and is denoted by (hat{mathfrak{a}}) symbolically (hat{mathfrak{a}}=frac{overrightarrow{mathrm{a}}}{|overrightarrow{mathrm{a}}|}.)

Equal Vectors  two vectors are said to be equal if they have the same magnitude, direction & represent the same physical quantity.

Collinear Vectors  two vectors are said to be collinear if their directed line segments are parallel disregards to their direction. Collinear vectors are also called Parallel Vectors. If they have the same direction they are named as like vectors otherwise unlike vectors. Symbolically, two non zero vectors (overrightarrow{mathrm{a}} text { and } overrightarrow{mathrm{b}}) are collinear if and only if, (overrightarrow{mathrm{a}}=mathrm{K} overrightarrow{mathrm{b}}), where K ∈ R Collinear Vectors a given number of vectors are called coplanar if their line segments are all parallel to the same plane. Note that “Two Vectors Are Always Coplanar”. Position Vector let O be a fixed origin, then the position vector of a point P is the vector (overrightarrow{mathrm{OP}}.)  If (overrightarrow{mathrm{a}quad}& overrightarrow{quad mathrm{b}})  & position vectors of two point A and B, then, (overrightarrow{mathrm{AB}}=overrightarrow{mathrm{b}}-overrightarrow{mathrm{a}}=mathrm{pv} text { of } mathrm{B}-mathrm{pv} text { of } mathrm{A})

Vector Addition | Vector Algebra

– If two vectors (overrightarrow{mathbf{a}} boldsymbol{boldsymbol { X }} overrightarrow{mathbf{b}})  are represented by (overrightarrow{O A} & overrightarrow{O B}) , then their sum (vec{a}+vec{b})  is a vector represented by (overrightarrow{mathrm{OC}}) where OC is the diagonal of the parallelogram OACB.
– (vec{a}+vec{b}=vec{b}+vec{a})  (commutative)
– ((vec{a}+vec{b})+vec{c}=vec{a}+(vec{b}+vec{c}))  (associativity)
–(overrightarrow{mathrm{a}}+overrightarrow{mathrm{O}}=overrightarrow{mathrm{a}}=overrightarrow{mathrm{O}}+overrightarrow{mathrm{a}})
– (overrightarrow{mathbf{a}}+(-overrightarrow{mathbf{a}})=overrightarrow{mathbf{O}}=(-overrightarrow{mathbf{a}})+overrightarrow{mathbf{a}})

Multiplication of Vector By Scalars | Vector Algebra

If (overrightarrow{mathrm{a}}) is a vector & m is a scalar, then (mathrm{m} overrightarrow{mathrm{a}}) is a vector parallel to (overrightarrow{mathrm{a}}) whose
modulus is |m| times that of (overrightarrow{mathrm{a}}.) This multiplication is called Scalar Multiplication. If (overrightarrow{mathrm{a}} & overrightarrow{mathrm{b}}) are vectors & m, n are scalars, then:
(begin{array}{l}{mathrm{m}(overrightarrow{mathrm{a}})=(overrightarrow{mathrm{a}}) mathrm{m}=mathrm{ma}} {mathrm{m}(mathrm{n} overrightarrow{mathrm{a}})=mathrm{n}(mathrm{m} overrightarrow{mathrm{a}})=(mathrm{mn}) overrightarrow{mathrm{a}}} {(mathrm{m}+mathrm{n}) overrightarrow{mathrm{a}}=mathrm{m} overrightarrow{mathrm{a}}+mathrm{n} overrightarrow{mathrm{a}}}end{array}) (m(vec{a}+vec{b})=m vec{a}+m vec{b})

Section Formula | Vector Algebra

If (overrightarrow{mathrm{a}} quad & overrightarrow{mathrm{b}}) are the position vectors of two points A & B then the p.v. of a point which divides AB in the ratio m : n is given by:
(vec{r}=frac{n vec{a}+m vec{b}}{m+n}.) Note p.v.
of mid point of AB (=frac{vec{a}+vec{b}}{2})

Direction Cosines | Vector Algebra

Let (vec{a}=a_{1} hat{i}+a_{2} hat{j}+a_{3} hat{k}) the angles which this vector makes with the +ve directions OX,OY & OZ are called Direction Angles & their cosines are called the Direction Cosines (cos alpha=frac{a_{1}}{|vec{a}|} quad, quad cos beta=frac{a_{2}}{|vec{a}|},cos Gamma=frac{a_{3}}{|vec{a}|}.)
Note that, cos² α + cos² β + cos² Γ = 1

Vector Equation of A Line | Vector Algebra

Parametric vector equation of a line passing through two point (mathrm{A}(overrightarrow{mathrm{a}}) & mathrm{B}(overrightarrow{mathrm{b}})) is given by, (overrightarrow{mathbf{r}}=overrightarrow{mathbf{a}}+t(vec{b}-overrightarrow{mathbf{a}})) where t is a parameter. If the line passes through the point (mathrm{A}(overrightarrow{mathrm{a}}))  & is parallel to the vector (overrightarrow{mathrm{b}}) then its equation is, (overrightarrow{mathrm{r}}=overrightarrow{mathrm{a}}+mathrm{t} overrightarrow{mathrm{b}}) Note that the equations of the bisectors of the angles between the lines (overrightarrow{mathrm{r}}=overrightarrow{mathrm{a}}+lambda overrightarrow{mathrm{b}}quad & quad overrightarrow{mathrm{r}}=overrightarrow{mathrm{a}}+mu overrightarrow{mathrm{c}}) is:
(overrightarrow{mathrm{r}}=overrightarrow{mathrm{a}}+mathrm{t}(hat{mathrm{b}}+hat{mathrm{c}}) quad & quad overrightarrow{mathrm{r}}=overrightarrow{mathrm{a}}+mathrm{p}(hat{mathrm{c}}-hat{mathrm{b}}))

Test of Collinearity

Three points A,B,C with position vectors (vec{a}, vec{b}, vec{c}) respectively are collinear, if & only if there exist scalars x, y , z not all zero simultaneously such that;
(x vec{a}+y vec{b}+z vec{c}=0) where x + y + z = 0.

Scalar Product Of Two Vectors

– (overrightarrow{mathrm{a}} cdot overrightarrow{mathrm{b}}=|overrightarrow{mathrm{a}}||overrightarrow{mathrm{b}}| cos theta(0 leq theta leq pi),) note that if θ is acute then (overrightarrow{mathrm{a}} overrightarrow{mathrm{b}}>0) & if θ is obtuse then (overrightarrow{mathrm{a}} overrightarrow{mathrm{b}}<0)
– (overrightarrow{mathrm{a}} overrightarrow{mathrm{a}}=|overrightarrow{mathrm{a}}|^{2}=overrightarrow{mathrm{a}}^{2}, overrightarrow{mathrm{a}} overrightarrow{mathrm{b}}=overrightarrow{mathrm{b}} overrightarrow{mathrm{a}}) (commutative)
– (vec{a} cdot(vec{b}+vec{c})=vec{a} cdot vec{b}+vec{a} cdot vec{c}) (distributive)
– (overrightarrow{mathrm{a}} cdot overrightarrow{mathrm{b}}=0 Leftrightarrow overrightarrow{mathrm{a}} perp overrightarrow{mathrm{b}} quad(overrightarrow{mathrm{a}} neq 0 quad overrightarrow{mathrm{b}} neq 0))
– (hat{mathrm{i}} . hat{mathrm{i}}=hat{mathrm{j}} cdot hat{mathrm{j}}=hat{mathrm{k}} cdot hat{mathrm{k}}=1 ; hat{mathrm{i}} cdot hat{mathrm{j}}=hat{mathrm{j}} hat{mathrm{k}}=hat{mathrm{k}} cdot hat{mathrm{i}}=0)
– projection of (overrightarrow{mathrm{a}} text { on } overrightarrow{mathrm{b}}=frac{overrightarrow{mathrm{a}} cdot overrightarrow{mathrm{b}}}{|overrightarrow{mathrm{b}}|})
Note: That vector component of (vec{a} text { along } vec{b}=left(frac{vec{a} cdot vec{b}}{vec{b}^{2}}right) vec{b}) and perpendicular to (overrightarrow{mathrm{b}}=overrightarrow{mathrm{a}}-left(frac{overrightarrow{mathrm{a}} cdot overrightarrow{mathrm{b}}}{overrightarrow{mathrm{b}}^{2}}right) overrightarrow{mathrm{b}}.)
–  the angle φ between (overrightarrow{mathrm{a}} quad &quad overrightarrow{mathrm{b}}) is given by
(cos phi=frac{overrightarrow{mathrm{a}} cdot overrightarrow{mathrm{b}}}{|overrightarrow{mathrm{a}}| overrightarrow{mathrm{b}} |})  0 ≤ φ ≤ π
– if (vec{a}=a_{1} hat{i}+a_{2} hat{j}+a_{3} hat{k} quad & quad vec{b}=b_{1} hat{i}+b_{2} hat{j}+b_{3} hat{k} text { then } vec{a} vec{b}=a_{1} b_{1}+a_{2} b_{2}+a_{3} b_{3})
(|overrightarrow{mathrm{a}}|=sqrt{mathrm{a}_{1}^{2}+mathrm{a}_{2}^{2}+mathrm{a}_{3}^{2}},)(|overrightarrow{mathrm{b}}|=sqrt{mathrm{b}_{1}^{2}+mathrm{b}_{2}^{2}+mathrm{b}_{3}^{2}})
Note:
(i) Maximum value of (overrightarrow{mathrm{a}} cdot overrightarrow{mathrm{b}}=|overrightarrow{mathrm{a}}||overrightarrow{mathrm{b}}|)
(ii) Minimum values of (vec{a} cdot vec{b}=vec{a} cdot vec{b}=-|vec{a}||vec{b}|)
(iii) Any vector (overrightarrow{mathrm{a}}) can be written as,(vec{a}=(vec{a} cdot hat{i}) hat{i}+(vec{a} cdot hat{j}) hat{j}+(vec{a} cdot hat{k}) hat{k})
(iv) A vector in the direction of the bisector of the angle between the two vectors (overrightarrow{mathrm{a}} & overrightarrow{mathrm{b}} text { is } frac{overrightarrow{mathrm{a}}}{|overrightarrow{mathrm{a}}|}+frac{overrightarrow{mathrm{b}}}{|overrightarrow{mathrm{b}}|}.) Hence bisector of the angle between the two vectors (overrightarrow{mathrm{a}} & overrightarrow{mathrm{b}} text { is } lambda(hat{mathrm{a}}+hat{mathrm{b}}),) where λ ∈ R+. Bisector of the exterior angle between (vec{a} & vec{b} text { is } lambda(hat{a}-hat{b}), lambda in R^{+}.)

Vector Product Of Two Vectors

• If (overrightarrow{mathrm{a}} & overrightarrow{mathrm{b}}) are two vectors & θ is the angle between them then [latexll]overrightarrow{mathbf{a}} times overrightarrow{mathbf{b}}=|overrightarrow{mathbf{a}}| vec{b} | sin theta overrightarrow{mathbf{n}}[/latex] where (overrightarrow{mathrm{n}}) is the unit vector perpendicular to both (vec{a} & vec{b}) such that [latexl]vec{a}, vec{b} & vec{n}l[/latex] forms a right handed screw system.
• Lagranges Identity : for any two vectors
  (vec{a} & vec{b} ;(vec{a} times vec{b})^{2}=left.|vec{a}|right|^{2}|vec{b}|^{2}-(vec{a} cdot vec{b})^{2}=left| begin{array}{ll}{vec{a} cdot vec{a}} & {vec{a} . vec{b}} {overrightarrow{mathbf{a}} cdot vec{b}} & {vec{b} . vec{b}}end{array}right|)
• Formulation of vector product in terms of scalar product: The vector product (overrightarrow{mathbf{a}} times overrightarrow{mathbf{b}}) is the vector (overrightarrow{mathrm{c}}) , such that
  (i) (|vec{c}|=sqrt{vec{a}^{2} vec{b}^{2}-(vec{a} cdot vec{b})^{2}})
  (ii) (overrightarrow{mathrm{c}} cdot overrightarrow{mathrm{a}}=0 ; quad overrightarrow{mathrm{c}} cdot overrightarrow{mathrm{b}}=0)  and
  (iii) (overrightarrow{mathrm{a}}, overrightarrow{mathrm{b}}, overrightarrow{mathrm{c}}) form a right handed system
• (overrightarrow{mathrm{a}} times overrightarrow{mathrm{b}}=0 Leftrightarrow overrightarrow{mathrm{a}} & overrightarrow{mathrm{b}}) are parallel (collinear) ((vec{a} neq mathbf{O}, overrightarrow{mathbf{b}} neq mathbf{O}) text { i.e. } overrightarrow{mathbf{a}}=mathbf{K} overrightarrow{mathbf{b}}) , where K is a scalar.
  – (overrightarrow{mathrm{a}} times overrightarrow{mathrm{b}} neq overrightarrow{mathrm{b}} times overrightarrow{mathrm{a}}) (not commutative)
  – ((mathrm{m} overrightarrow{mathrm{a}}) times overrightarrow{mathrm{b}}=overrightarrow{mathrm{a}} times(mathrm{m} overrightarrow{mathrm{b}})=mathrm{m}(overrightarrow{mathrm{a}} times overrightarrow{mathrm{b}})) where m is a scalar.
  – (overrightarrow{mathbf{a}} times(vec{b}+vec{c})=(vec{a} times vec{b})+(vec{a} times vec{c})) (distributive)
  – (hat{mathrm{i}} times hat{mathrm{i}}=hat{mathrm{j}} times hat{mathrm{j}}=hat{mathrm{k}} times hat{mathrm{k}}=0)
  – (hat{mathrm{i}} times hat{mathrm{j}}=hat{mathrm{k}}, hat{mathrm{j}} times hat{mathrm{k}}=hat{mathrm{i}}, hat{mathrm{k}} times hat{mathrm{i}}=hat{mathrm{j}})
• If
  (overrightarrow{mathrm{a}}=mathrm{a}_{1} hat{mathrm{i}}+mathrm{a}_{2} hat{mathrm{j}}+mathrm{a}_{3} hat{mathrm{d}} quad & quad overrightarrow{mathrm{b}}=mathrm{b}_{1} hat{mathrm{i}}+mathrm{b}_{2} hat{mathrm{j}}+mathrm{b}_{3} hat{mathrm{k}}text {then}overrightarrow{mathrm{a}} times overrightarrow{mathrm{b}}=left| begin{array}{ccc}{hat{mathbf{i}}} & {hat{mathbf{j}}} & {hat{mathbf{k}}} {mathrm{a}_{1}} & {mathrm{a}_{2}} & {mathrm{a}_{3}} {mathrm{b}_{1}} & {mathrm{b}_{2}} & {mathrm{b}_{3}}end{array}right|)
• Geometrically (|overrightarrow{mathrm{a}} times overrightarrow{mathrm{b}}|) = area of the parallelogram whose two adjacent sides are represented by (overrightarrow{mathbf{a}} & overrightarrow{mathbf{b}})
• Unit vector perpendicular to the plane of (overrightarrow{mathrm{a}} & overrightarrow{mathrm{b}} text { is } hat{mathrm{n}}=pm frac{overrightarrow{mathrm{a}} times overrightarrow{mathrm{b}}}{|overrightarrow{mathrm{a}} times overrightarrow{mathrm{b}}|})
  –  A vector of magnitude ‘r’ & perpendicular to the palne of (overrightarrow{mathrm{a}} & overrightarrow{mathrm{b}} text { is } pm frac{mathrm{r}(overrightarrow{mathrm{a}} times overrightarrow{mathrm{b}})}{|overrightarrow{mathrm{a}} times overrightarrow{mathrm{b}}|})
  –  If θ is the angle between (overrightarrow{mathrm{a}} & overrightarrow{mathrm{b}} text { then sin } theta=frac{|overrightarrow{mathrm{a}} times overrightarrow{mathrm{b}}|}{|overrightarrow{mathrm{a}}||overrightarrow{mathrm{b}}|})
• Vector area
  –  If (vec{a}, vec{b} & vec{c}) are the pv’s of 3 points A, B & C then the vector area of triangle ABC
  (=frac{1}{2}[overrightarrow{mathrm{a}} mathrm{x} overrightarrow{mathrm{b}}+overrightarrow{mathrm{b}} mathrm{x} overrightarrow{mathrm{c}}+overrightarrow{mathrm{c}} mathrm{x} overrightarrow{mathrm{a}}].) The points A, B & C are collinear if (overrightarrow{mathrm{a}} times overrightarrow{mathrm{b}}+overrightarrow{mathrm{b}} mathrm{x} overrightarrow{mathrm{c}}+overrightarrow{mathrm{c}} mathrm{x} overrightarrow{mathrm{a}}=0)
  –  Area of any quadrilateral whose diagonal vectors are (overrightarrow{mathrm{d}}_{1} & overrightarrow{mathrm{d}}_{2}) is given by (frac{1}{2}left|overrightarrow{mathrm{d}}_{1} mathrm{x} overrightarrow{mathrm{d}}_{2}right|)

Shortest Distance Between Two Lines

If two lines in space intersect at a point, then obviously the shortest distance between them is zero. Lines which do not intersect & are also not parallel are called Skew lines. For Skew lines the direction of the shortest distance would be perpendicular to both the lines. The magnitude of the shortest distance vector would be equal to that of the projection of (overrightarrow{mathrm{AB}}) along the direction of the line of shortest distance, (overrightarrow{mathrm{L} mathrm{M}}) is parallel to (overrightarrow{mathbf{p}} times overrightarrow{mathbf{q}}) i. e.
(overrightarrow{mathrm{LM}}=| text { Projection of } overrightarrow{mathrm{AB}} text { on } mathrm{LM}|=| text { Projection of } overrightarrow{mathrm{AB}} text { on } overrightarrow{mathrm{p}} times overline{mathrm{q}}|=left|frac{overrightarrow{mathrm{AB}} cdot(overrightarrow{mathrm{p}} times overrightarrow{mathrm{q}})}{overrightarrow{mathrm{p}} times overrightarrow{mathrm{q}}}right|=left|frac{(overrightarrow{mathrm{b}}-overrightarrow{mathrm{a}}) cdot(overrightarrow{mathrm{p}} mathrm{x} overrightarrow{mathrm{q}})}{|overrightarrow{mathrm{p}} times overrightarrow{mathrm{q}}|}right|)

1. The two lines directed along (overrightarrow{mathrm{p}} & overrightarrow{mathrm{q}}) will intersect only if shortest distance = 0 i.e. ((vec{b}-vec{a}) cdot(vec{p} x vec{q})=0 quad text { i.e. }(vec{b}-vec{a})) lies in the plane containing (overrightarrow{mathrm{p}} & overrightarrow{mathrm{q}} rightarrow[(overrightarrow{mathrm{b}}-overrightarrow{mathrm{a}}) overrightarrow{mathrm{p}} mathrm{q}]=0)
2. If two lines are given by (overrightarrow{mathrm{r}}_{1}=overrightarrow{mathrm{a}}_{1}+mathrm{K} overrightarrow{mathrm{b}} & overrightarrow{mathrm{r}}_{2}=overrightarrow{mathrm{a}}_{2}+mathrm{K} overrightarrow{mathrm{b}}) i.e. they are parallel then,
(mathrm{d}=left|frac{overrightarrow{mathrm{b}} mathrm{x}left(overrightarrow{mathrm{a}}_{2}-overrightarrow{mathrm{a}}_{1}right)}{|overrightarrow{mathrm{b}}|}right|)

Scalar Triple Product / Box Product / Mixed Product

– The scalar triple product of three vectors (overrightarrow{mathbf{a}}, overrightarrow{mathbf{b}} & overrightarrow{mathbf{c}}) is defined as:
(overrightarrow{mathbf{a}} times overrightarrow{mathrm{b}} overrightarrow{mathrm{c}}=|overrightarrow{mathrm{a}}| overrightarrow{mathrm{b}}|overrightarrow{mathrm{c}}|) sin θ cos φ where θ is the angle between (overrightarrow{mathrm{a}} & overrightarrow{mathrm{b}}&phi) is the angle between (overrightarrow{mathrm{a}} times overrightarrow{mathrm{b}} & overrightarrow{mathrm{c}}.) It is also defined as ([vec{a} vec{b} vec{c}]) , spelled as box product.
– Scalar triple product geometrically represents the volume ofthe parallelopiped whose three couterminous
edges are represented by
(overrightarrow{mathbf{a}}, overrightarrow{mathbf{b}} & overrightarrow{mathbf{c}} mathbf{i} . mathbf{e} cdot mathbf{V}=[overrightarrow{mathbf{a}} overrightarrow{mathbf{b}} overrightarrow{mathbf{c}}])
– In a scalar triple product the position of dot & cross can be interchanged i.e.
(overrightarrow{mathbf{a}} cdot(overrightarrow{mathbf{b}} mathbf{x} overrightarrow{mathbf{c}})=(overrightarrow{mathbf{a}} mathbf{x} overrightarrow{mathbf{b}}) overrightarrow{mathbf{c}} text {OR}[overrightarrow{mathbf{a}} overrightarrow{mathbf{b}} overrightarrow{mathbf{c}}]=[overrightarrow{mathbf{b}} overrightarrow{mathbf{a}} mathbf{a}]=[overrightarrow{mathbf{c}} overrightarrow{mathbf{a}} overrightarrow{mathbf{b}}])
– (overrightarrow{mathrm{a}} cdot(overrightarrow{mathrm{b}} mathrm{x} overrightarrow{mathrm{c}})=-overrightarrow{mathrm{a}} cdot(overrightarrow{mathrm{c}} mathrm{x} overrightarrow{mathrm{b}}) text { i.e. }[overrightarrow{mathrm{a}} overrightarrow{mathrm{b}} overrightarrow{mathrm{c}}]=-[overrightarrow{mathrm{a}} overrightarrow{mathrm{c}} overrightarrow{mathrm{b}}])
– If (vec{a}=a_{1} hat{i}+a_{2} hat{j}+a_{3} hat{k} quad ; vec{b}=b_{1} hat{i}+b_{2} hat{j}+b_{3} hat{k} quad & vec{c}=c_{1} hat{i}+c_{2} hat{j}+c_{3} hat{k}) then.
In general, if
(overrightarrow{mathrm{a}}=mathrm{a}_{1} overrightarrow{mathrm{l}}+mathrm{a}_{2} overrightarrow{mathrm{m}}+mathrm{a}_{3} overrightarrow{mathrm{n}} quad ; quad overrightarrow{mathrm{b}}=mathrm{b}_{1} overrightarrow{mathrm{l}}+mathrm{b}_{2} overrightarrow{mathrm{m}}+mathrm{b}_{3} overrightarrow{mathrm{n}} quad & quad overrightarrow{mathrm{c}}=mathrm{c}_{1} overrightarrow{mathrm{l}}+mathrm{c}_{2} overrightarrow{mathrm{m}}+mathrm{c}_{3} overrightarrow{mathrm{n}})
then
([overrightarrow{mathrm{a}} overrightarrow{mathrm{b}} overrightarrow{mathrm{c}}]=left| begin{array}{lll}{mathrm{a}_{1}} & {mathrm{a}_{2}} & {mathrm{a}_{3}} {mathrm{b}_{1}} & {mathrm{b}_{2}} & {mathrm{b}_{3}} {mathrm{c}_{1}} & {mathrm{c}_{2}} & {mathrm{c}_{3}}end{array}right|[overrightarrow{1} mathrm{m} overrightarrow{mathrm{n}}])  where (vec{ell}, overrightarrow{mathrm{m}} & overrightarrow{mathrm{n}}) are non coplanar vectors.
–  If (overrightarrow{mathbf{a}}, overrightarrow{mathbf{b}}, overrightarrow{mathbf{c}}) are coplanar ⇔ ([vec{a} vec{b} vec{c}]=0.)
Note: If (overrightarrow{mathrm{a}}, overrightarrow{mathrm{b}}, overrightarrow{mathrm{c}}) are non − coplanar then ([vec{a} vec{b} vec{c}]>0) for right handed system & ([vec{a} vec{b} vec{c}]<0) for left handed system
– [i j k] = 1
– ([mathrm{Ka} overrightarrow{mathrm{b}} overrightarrow{mathrm{c}}]=mathrm{K}[overrightarrow{mathrm{a}} overrightarrow{mathrm{b}} overrightarrow{mathrm{c}}])
– ([(vec{a}+vec{b}) vec{c} vec{d}]=[vec{a} vec{c} vec{d}]+[vec{b} vec{c} vec{d}])
–  he volume ofthe tetrahedron OABC with O as origin & the pv’s ofA, B and C being (overrightarrow{mathrm{a}}, overrightarrow{mathrm{b}} & overrightarrow{mathrm{c}}) respectively is given by (mathrm{V}=frac{1}{6}[overrightarrow{mathrm{a}} overrightarrow{mathrm{b}} overrightarrow{mathrm{c}}])
–  The positon vector of the centroid of a tetrahedron if the pv’s of its angular vertices are (overrightarrow{mathbf{a}}, overrightarrow{mathbf{b}}, overrightarrow{mathbf{c}} & overrightarrow{mathbf{d}}) are given by (frac{1}{4}[vec{a}+vec{b}+vec{c}+vec{d}])
Note that this is also the point of concurrency of the lines joining the vertices to the centroids of the opposite faces and is also called the centre of the tetrahedron. In case the tetrahedron is regular it is equidistant from the vertices and the four faces of the tetrahedron .
Remember that:
(left[ begin{array}{ccc}{vec{a}-vec{b}} & {vec{b}-vec{c}} & {vec{c}-vec{a}}end{array}right]=0 quad & quad[vec{a}+vec{b} quad vec{b}+vec{c} quad vec{c}+vec{a}]=2[vec{a} vec{b} vec{c}])

Vector Triple Product

Let (overrightarrow{mathbf{a}}, overrightarrow{mathbf{b}}, overrightarrow{mathbf{c}}) be any three vectors, then the expression (overrightarrow{mathbf{a}} times(overrightarrow{mathbf{b}} times overrightarrow{mathbf{c}})) is a vector & is called a vector triple product.
Geometrical Interpretation Of(overrightarrow{mathbf{a}} times(overrightarrow{mathbf{b}} times overrightarrow{mathbf{c}}))
Consider the expression (overrightarrow{mathbf{a}} times(overrightarrow{mathbf{b}} times overrightarrow{mathbf{c}})) which itself is a vector, since it is a cross product of two vectors (vec{a} &(vec{b} x vec{c}) . text { Now } vec{a} times(vec{b} times vec{c})) is a vector perpendicular to the plane containing (vec{a} &(vec{b} x vec{c})) but (overrightarrow{mathrm{b}} times overrightarrow{mathrm{c}}l) is a
vector perpendicular to the plane (overrightarrow{mathrm{b}} & overrightarrow{mathrm{c}},) therefore (overrightarrow{mathbf{a}} times(vec{b} times overrightarrow{mathbf{c}})) is a vector lies in the plane of (overrightarrow{mathrm{b}} & overrightarrow{mathrm{c}}) and perpendicular to (overrightarrow{mathrm{a}}.) Hence we can express [/latex]overrightarrow{mathbf{a}} times(overrightarrow{mathbf{b}} mathbf{x} overrightarrow{mathbf{c}})[/latex] in terms of (overrightarrow{mathrm{b}} & overrightarrow{mathrm{c}}) i.e. (vec{a} times(vec{b} times vec{c})=x vec{b}+y vec{c}) where x & y are scalars.
– (overrightarrow{mathbf{a}} times(overrightarrow{mathbf{b}} times overrightarrow{mathbf{c}})=(vec{a} cdot overrightarrow{mathbf{c}}) overrightarrow{mathbf{b}}-(overrightarrow{mathbf{a}} cdot overrightarrow{mathbf{b}}) overrightarrow{mathbf{c}})
– ((vec{a} times vec{b}) times vec{c}=(vec{a} cdot vec{c}) vec{b}-(vec{b} cdot vec{c}) vec{a})
– ((vec{a} times vec{b}) times vec{c} neq vec{a} times(vec{b} times vec{c}))

Linear Combinations / Linearly Independence and Dependence of Vectors

Given a finite set of vectors (overrightarrow{mathbf{a}}, overrightarrow{mathbf{b}}, overrightarrow{mathbf{c}}, ldots) then the vector (overrightarrow{mathbf{r}}=mathbf{x} overrightarrow{mathbf{a}}+mathbf{y} overrightarrow{mathbf{b}}+mathbf{z} overrightarrow{mathbf{c}}+ldots ldots ldots) is called a linear
combination of (overrightarrow{mathrm{a}}, overrightarrow{mathrm{b}}, overrightarrow{mathrm{c}}, ldots ldots) for any x, y, z …… ∈ R. We have the following results:

• Fundamentaltheorem In Plane:  Let (vec{a}, vec{b}) be non zero , non collinear vectors . Then any vector (overrightarrow{mathrm{r}}) coplanar with (overrightarrow{mathbf{a}}, overrightarrow{mathbf{b}})  can be expressed uniquely as a linear combination of (overrightarrow{mathrm{a}}, overrightarrow{mathrm{b}})  i.e. There exist some unique x ,y ∈ R such that (x vec{a}+y vec{b}=vec{r})
• Fundamental Theorem In Space: Let (overrightarrow{mathbf{a}}, overrightarrow{mathbf{b}}, overrightarrow{mathbf{c}}) be non−zero, non−coplanar vectors in space. Then any vector (overrightarrow{mathrm{r}},) can be uniquily expressed as a linear combination of (overrightarrow{mathbf{a}}, overrightarrow{mathbf{b}} overrightarrow{mathbf{c}}) i.e. There exist some unique x,y ∈ R such that (x vec{a}+y vec{b}+z vec{c}=vec{r}.)
• If (overrightarrow{mathbf{x}}_{1}, overrightarrow{mathbf{x}}_{2}, ldots ldots overrightarrow{mathbf{x}}_{mathbf{n}}) are n non zero vectors, & k₁, k₂, …..k_n are n scalars & if the linear combination
  (k_{1} vec{x}_{1}+k_{2} vec{x}_{2}+ldots ldots k_{n} vec{x}_{n}=0 Rightarrow k_{1}=0, k_{2}=0 ldots k_{n}=0) then we say that vectors (vec{x}_{1}, vec{x}_{2}, ldots . . vec{x}_{n}) are
  Linearly Independent Vectors.
• If (vec{x}_{1}, vec{x}_{2}, ldots . . vec{x}_{n}) Linearly Independent then they are said to be Linearly Dependent vectors i.e. if
  (mathbf{k}_{1} overrightarrow{mathbf{x}}_{1}+mathbf{k}_{2} overrightarrow{mathbf{x}}_{2}+ldots ldots+mathbf{k}_{mathbf{n}} overrightarrow{mathbf{x}}_{mathbf{n}}=mathbf{0}) & if there exists at least one k_r≠ 0 then (overrightarrow{mathbf{x}}_{1}, overrightarrow{mathbf{x}}_{2}, ldots ldots overrightarrow{mathbf{x}}_{mathbf{n}}) are said to be Linearly Dependent.
  Note:
  –  If (vec{a}=3 i+2 j+5 k text { then } vec{a}) is expressed as a Linear Combination of vectors (hat{mathrm{i}}, hat{mathrm{j}}, hat{mathrm{k}}.) Also, (overrightarrow{mathrm{a}}, hat{mathrm{i}}, hat{mathrm{j}}, hat{mathrm{j}}) form a linearly dependent set of vectors. In general , every set of four vectors is a linearly dependent system.
  – (hat{mathbf{i}}, hat{mathbf{j}}, hat{mathbf{k}}) are Linearly Independent set of vectors. For
  (mathrm{K}_{1} hat{mathrm{i}}+mathrm{K}_{2} hat{mathrm{j}}+mathrm{K}_{3} hat{mathrm{k}}=0 Rightarrow mathrm{K}_{1}=0=mathrm{K}_{2}=mathrm{K}_{3})
  –  Two vectors (overrightarrow{mathrm{a}} & overrightarrow{mathrm{b}}) are linearly dependent ⇒ (overrightarrow{mathrm{a}}) is parallel to (vec{b} text { i.e. } vec{a} times vec{b}=0 Rightarrow) linear dependence of (overrightarrow{mathrm{a}} & overrightarrow{mathrm{b}}.) Conversely if (vec{a} times vec{b} neq 0 text { then } vec{a} & vec{b}) are linearly independent.
  –  If three vectors (overrightarrow{mathbf{a}}, overrightarrow{mathbf{b}}, overrightarrow{mathbf{c}}) are linearly dependent, then they are coplanar i.e. ([vec{a}, vec{b}, vec{c}]=0,) conversely, if ([vec{a}, vec{b}, vec{c}] neq 0) , then the vectors are linearly independent.

Coplanarity Of Vectors

Four points A, B, C, D with position vectors (vec{a}, vec{b}, vec{c}, vec{d}) respectively are coplanar if and only if there exist scalars x, y, z, w not all zero simultaneously such that (x vec{a}+y vec{b}+z vec{c}+w vec{d}=0) where, x + y + z + w = 0.

Reciprocal System Of Vectors

If (overrightarrow{mathbf{a}}, overrightarrow{mathbf{b}}, overrightarrow{mathbf{c}} quad & vec{a}^{prime} vec{b}^{prime} vec{c}^{prime}) are two sets of non coplanar vectors such that (vec{a} cdot vec{a}^{prime}=vec{b} cdot vec{b}^{prime}=vec{c} cdot vec{c}^{prime}=1) then the two systems are called Reciprocal System of vectors.
Note:(a^{prime}=frac{vec{b} x vec{c}}{[vec{a} vec{b} vec{c}]} ; b^{prime}=frac{vec{c} x vec{a}}{[vec{a} vec{b} vec{c}]} ; c^{prime}=frac{vec{a} times vec{b}}{[vec{a} vec{b} vec{c}]})

Equation Of A Plane

• The equation (left(vec{r}-vec{r}_{0}right) vec{n}=0) represents a plane containing the point with p.v.(overrightarrow{mathrm{r}}_{0}) where (overrightarrow{mathrm{n}}) is a vector normal to the plane.
  (overrightarrow{mathbf{r}} cdot overline{mathbf{n}}=d) is the general equation of a plane.
• Angle between the 2 planes is the angle between 2 normals drawn to the planes and the angle between a line and a plane is the compliment of the angle between the line and the normal to the plane.

Application of Vectors

1. Work done against a constant force (overrightarrow{mathrm{F}}) over adisplacement (overrightarrow{mathbf{s}}) is defined as (overrightarrow{mathrm{W}}=overrightarrow{mathrm{F}} overrightarrow{mathrm{s}})
   
2. The tangential velocity (overrightarrow{mathrm{V}}) of a body moving in a circle is given by (overrightarrow{mathrm{V}}=overrightarrow{mathrm{w}} times overrightarrow{mathrm{r}} text { where } overrightarrow{mathrm{r}}) is the pv of the point P
3. The moment of (overrightarrow{mathrm{F}}) about ’O’ is defined as (overrightarrow{mathrm{M}}=overrightarrow{mathrm{r}} times overrightarrow{mathrm{F}}) where (overrightarrow{mathbf{r}}) is the pv of P wrt ’O’. The direction of (overrightarrow{mathbf{M}}) is along the normal to the plane OPN such that (overrightarrow{mathbf{r}}, overrightarrow{mathbf{F}} boldsymbol{&} mathbf{vec { M }}) form a right handed system.
   
4. Moment of the couple (=left(overrightarrow{mathrm{r}}_{1}-overrightarrow{mathrm{r}}_{2}right) times overrightarrow{mathrm{F}} text { where } overrightarrow{mathrm{r}}_{1} & overrightarrow{mathrm{r}}_{2}) are pv’s of the point of the application of the forces (overrightarrow{mathbf{F}} boldsymbol{boldsymbol { X }}-overrightarrow{mathbf{F}})

3 -D Coordinate Geometry | Vector Algebra

Distance (d) between two points (x₁ , y₁ , z₁) and (x₂ , y₂ , z₂)
(d=sqrt{left(x_{2}-x_{1}right)^{2}+left(y_{2}-y_{1}right)^{2}+left(z_{2}-z_{1}right)^{2}})

Section Formula:

(x=frac{m_{2} x_{1}+m_{1} x_{2}}{m_{1}+m_{2}} quad ; quad y=frac{m_{2} y_{1}+m_{1} y_{2}}{m_{1}+m_{2}} ; z=frac{m_{2} z_{1}+m_{1} z_{2}}{m_{1}+m_{2}}) ( For external division take –ve sign )
Direction Cosine and direction ratio’s of a line

Direction cosine of a line has the same meaning as d.c’s of a vector.

(a) Any three numbers a, b, c proportional to the direction cosines are called the direction ratios i.e.
(frac{l}{mathrm{a}}=frac{mathrm{m}}{mathrm{b}}=frac{mathrm{n}}{mathrm{c}}=pm frac{1}{sqrt{mathrm{a}^{2}+mathrm{b}^{2}+mathrm{c}^{2}}})
same sign either +ve or –ve should be taken through out.
note that d.r’s of a line joining x₁ , y₁ , z₁ and x₂ , y₂ , z₂ are proportional to x₂ – x₁ , y₂ – y₁ and z₂ – z₁
(b) If θ is the angle between the two lines whose d.c’s are l₁ , m₁ , n₁ and l₂ , m₂ , n₂
cosθ = l₁l₂ + m₁m₂+n₁n₂ hence if lines are perpendicular then l₁l₂ + m₁m₂+ n₁n₂ = 0 if lines are parallel then (frac{l_{1}}{l_{2}}=frac{mathrm{m}_{1}}{mathrm{m}_{2}}=frac{mathrm{n}_{1}}{mathrm{n}_{2}})
note that if three lines are coplanar then
(left| begin{array}{lll}{l_{1}} & {mathrm{m}_{1}} & {mathrm{n}_{1}} {l_{2}} & {mathrm{m}_{2}} & {mathrm{n}_{2}} {l_{3}} & {mathrm{m}_{3}} & {mathrm{n}_{3}}end{array}right|=0)

Projection of join of 2 points on line with d.c’s l, m, n are l (x₂ – x₁) + m(y₂ – y₁) + n(z₂ – z₁)
B. Plane

1. General equation of degree one in x, y, z i.e. ax + by + cz + d = 0 represents a plane.
2. Equation of a plane passing through (x₁ , y₁ , z₁) is a (x – x₁) + b (y – y₁) + c (z – z₁) = 0 where a, b, c are the direction ratios of the normal to the plane.
3. Equation of a plane if its intercepts on the co-ordinate axes are x₁ , y₁ , z₁ is
   (frac{x}{x_{1}}+frac{y}{y_{1}}+frac{z}{z_{1}}=1)
4. Equation of a plane if the length of the perpendicular from the origin on the plane is p and d.c’s of the perpendicular as l , m, , n is lx + my + nz = p
5. Parallel and perpendicular planes – Two planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 are perpendicular if a₁a₂ + b₁b₂ + c₁c₂ = 0  parallel if (frac{mathrm{a}_{1}}{mathrm{a}_{2}}=frac{mathrm{b}_{1}}{mathrm{b}_{2}}=frac{mathrm{c}_{1}}{mathrm{c}_{2}}) and coincident if (frac{mathrm{a}_{1}}{mathrm{a}_{2}}=frac{mathrm{b}_{1}}{mathrm{b}_{2}}=frac{mathrm{c}_{1}}{mathrm{c}_{2}}=frac{mathrm{d}_{1}}{mathrm{d}_{2}})
6. Angle between a plane and a line is the compliment of the angle between the normal to the plane and the
   line . If (left.begin{array}{l}{text { Line } : overrightarrow{mathrm{r}}=overrightarrow{mathrm{a}}+lambda overrightarrow{mathrm{b}}} {text { Plane } : overrightarrow{mathrm{r}} cdot overrightarrow{mathrm{n}}=mathrm{d}}end{array}right] text { then } cos (90-theta)=sin theta=frac{overrightarrow{mathrm{b}} cdot overrightarrow{mathrm{n}}}{|overrightarrow{mathrm{b}}| .|overrightarrow{mathrm{n}}|})
   where θ is the angle between the line and normal to the plane.
7. Length of the perpendicular from a point (x₁ , y₁ , z₁) to a plane ax + by + cz + d = 0 is
   (p=left|frac{a x_{1}+b y_{1}+c z_{1}+d}{sqrt{a^{2}+b^{2}+c^{2}}}right|)
8. Distance between two parallel planes ax + by + cz + d₁ = 0 and ax+by + cz + d₂ = 0 is
   (left|frac{mathrm{d}_{1}-mathrm{d}_{2}}{sqrt{mathrm{a}^{2}+mathrm{b}^{2}+mathrm{c}^{2}}}right|)
9. Planes bisecting the angle between two planes a₁x + b₁y + c₁z + d₁ = 0 and a₂ + b₂y + c₂z + d₂ = 0 is  given by
   (left|frac{a_{1} x+b_{1} y+c_{1} z+d_{1}}{sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}right|=pmleft|frac{a_{2} x+b_{2} y+c_{2} z+d_{2}}{sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}right|)
   Of these two bisecting planes , one bisects the acute and the other obtuse angle between the given planes.
10. Equation of a plane through the intersection of two planes P₁ and P₂is given by P₁+λP₂=0

Straight Line In Space

1. Equation of a line through A (x₁ , y₁ , z₁) and having direction cosines l ,m , n are
   (frac{mathrm{x}-mathrm{x}_{1}}{l}=frac{mathrm{y}-mathrm{y}_{1}}{mathrm{m}}=frac{mathrm{z}-mathrm{z}_{1}}{mathrm{n}}) and the lines through (x₁ , y₁ ,z₁) and (x₂ , y₂ ,z₂)
   (frac{x-x_{1}}{x_{2}-x_{1}}=frac{y-y_{1}}{y_{2}-y_{1}}=frac{z-z_{1}}{z_{2}-z_{1}})
2. Intersection of two planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 together represent the unsymmetrical form of the straight line.
3. General equation of the plane containing the line
   (frac{mathrm{x}-mathrm{x}_{1}}{l}=frac{mathrm{y}-mathrm{y}_{1}}{mathrm{m}}=frac{mathrm{z}-mathrm{z}_{1}}{mathrm{n}}) is  A (x – x₁) + B(y – y₁) + c (z – z₁) = 0 where Al + bm + cn = 0 .

Line of Greatest Slope
AB is the line of intersection of G-plane and H is the horizontal plane. Line ofgreatest slope on a given plane, drawn through a given point on the plane, is the line through the point ‘P’ perpendicular to the line of intersection of the given plane with any horizontal plane.

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