## NCERT Notes for Class 12 Mathematics

Chapter 10:
Vector Algebra

A **vector** has direction and magnitude both but scalar has only magnitude.

Magnitude of a vector a is denoted by |a| or a. It is non-negative scalar.

### Equality of Vectors

Two vectors a and b are said to be equal written as a = b, if they have (i) same length (ii) the
same or parallel support and (iii) the same sense.

### Types of Vectors

##### (i) Zero or Null Vector

A vector whose initial and terminal points are coincident is called zero
or null vector. It is denoted by 0.

##### (ii) Unit Vector

A vector whose magnitude is unity is called a unit vector which is denoted by
nˆ

##### (iii) Free Vectors

If the initial point of a vector is not specified, then it is said to be a free
vector.

##### (iv) Negative of a Vector

A vector having the same magnitude as that of a given vector a and
the direction opposite to that of a is called the negative of a and it is denoted by —a.

##### (v) Like and Unlike Vectors

Vectors are said to be like when they have the same direction and
unlike when they have opposite direction.

##### (vi) Collinear or Parallel Vectors

Vectors having the same or parallel supports are called
collinear vectors.

##### (vii) Coinitial Vectors

Vectors having same initial point are called coinitial vectors.

##### (viii) Coterminous Vectors

Vectors having the same terminal point are called coterminous
vectors.

##### (ix) Localized Vectors

A vector which is drawn parallel to a given vector through a specified
point in space is called localized vector.

##### (x) Coplanar Vectors

A system of vectors is said to be coplanar, if their supports are parallel
to the same plane. Otherwise they are called non-coplanar vectors.

##### (xi) Reciprocal of a Vector

A vector having the same direction as that of a given vector but
magnitude equal to the reciprocal of the given vector is known as the reciprocal of a. i.e., if |a| = a, then |a^{-1}| = 1 / a.

### Addition of Vectors

Let** a** and **b** be any two vectors. From the terminal point of a, vector b is drawn. Then, the
vector from the initial point O of a to the terminal point B of b is called the sum of vectors a
and b and is denoted by **a + b**. This is called the triangle law of addition of vectors.

#### Parallelogram Law

Let a and b be any two vectors. From the initial point of a, vector b is drawn and parallelogram
OACB is completed with OA and OB as adjacent sides. The vector OC is efined as the sum of
a and b. This is called the parallelogram law of addition of vectors.

The sum of two vectors is also called their resultant and the process of addition as composition.

#### Properties of Vector Addition

(i) a + b = b + a (commutativity)

(ii) a + (b + c)= (a + b)+ c (associativity)

(iii) a+ O = a (additive identity)

(iv) a + (— a) = 0 (additive inverse)

(v) (k_{1} + k_{2}) a = k_{1} a + k_{2}a (multiplication by scalars)

(vi) k(a + b) = k a + k b (multiplication by scalars)

(vii) |a+ b| ≤ |a| + |b| and |a – b| ≥ |a| – |b|

#### Difference (Subtraction) of Vectors

If a and b be any two vectors, then their difference a – b is defined as a + (- b).

#### Multiplication of a Vector by a Scalar

Let a be a given vector and λ be a scalar. Then, the product of the vector a by the scalar λ is λ a
and is called the multiplication of vector by the scalar.

##### Important Properties

(i) |λ a| = |λ| |a|

(ii) λ O = O

(iii) m (-a) = – ma = – (m a)

(iv) (-m) (-a) = m a

(v) m (n a) = mn a = n(m a)

(vi) (m + n)a = m a+ n a

(vii) m (a+b) = m a + m b

##### Vector Equation of Joining by Two Points

Let P_{1} (x_{1}, y_{1}, z_{1}) and P_{2} (x_{2}, y_{2}, z_{2}) are any two points, then the vector joining P_{1} and P_{2} is the
vector P_{1} P_{2}.

The component vectors of P and Q are

OP = x_{1}i + y_{1}j + z_{1}k and OQ = x_{2}i + y_{2}j + z_{2}k

i.e., P_{1} P_{2} = (x_{2}i + y_{2}j + z_{2}k) – (x_{1}i + y_{1}j + z_{1}k)

= (x_{2} – x_{1}) i + (y_{2} – y_{1}) j + (z_{2} – z_{1}) k

Its magnitude is P_{1} P_{2} = √(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2} + (z_{2} – z_{1})^{2}

##### Position Vector of a Point

The position vector of a point P with respect to a fixed point, say O, is the vector OP. The fixed
point is called the origin.

Let PQ be any vector. We have PQ = PO + OQ = — OP + OQ = OQ — OP = Position vector
of Q — Position vector of P.

i.e., PQ = PV of Q — PV of P

##### Collinear Vectors

Vectors a and b are collinear, if a = λb, for some non-zero scalar λ.

##### Collinear Points

Let A, B, C be any three points.

Points A, B, C are collinear <=> AB, BC are collinear vectors.

<=> AB = λBC for some non-zero scalar λ.

##### Section Formula

Let A and B be two points with position vectors a and b, respectively and OP= r.

(i) Let P be a point dividing AB internally in the ratio m : n. Then,

r = m b + n a / m + n

Also, (m + n) OP = m OB + n OA

(ii) The position vector of the mid-point of a and b is a + b / 2.

(iii) Let P be a point dividing AB externally in the ratio m : n. Then, r = m b + n a / m + n

#### Position Vector of Different Centre of a Triangle

(i) If a, b, c be PV’s of the vertices A, B, C of a ΔABC respectively, then the PV of the centroid
G of the triangle is a + b + c / 3.

(ii) The PV of incentre of ΔABC is (BC)a + (CA)b + (AB)c / BC + CA + AB

(iii) The PV of orthocentre of ΔABC is
a(tan A) + b(tan B) + c(tan C) / tan A + tan B + tan C

#### Scalar Product of Two Vectors

If a and b are two non-zero vectors, then the scalar or dot product of a and b is denoted by a * b
and is defined as a * b = |a| |b| cos θ, where θ is the angle between the two vectors and 0 < θ < π
.

(i) The angle between two vectors a and b is defined as the smaller angle θ between them,
when they are drawn with the same initial point. Usually, we take 0 < θ < π.Angle between two like vectors is O and angle between two unlike
vectors is π .

(ii) If either a or b is the null vector, then scalar product of the vector is zero.

(iii) If a and b are two unit vectors, then a * b = cos θ.

(iv) The scalar product is commutative i.e., a * b= b * a

(v) If i , j and k are mutually perpendicular unit vectors i , j and k, then
i * i = j * j = k * k =1 and i * j = j * k = k * i = 0

(vi) The scalar product of vectors is distributive over vector addition.

(a) a * (b + c) = a * b + a * c (left distributive)

(b) (b + c) * a = b * a + c * a (right distributive)

Note Length of a vector as a scalar product If a be any vector, then the scalar product a * a = |a| |a| cosθ ⇒ |a|2 = a2 ⇒ a = |a| Condition of perpendicularity a * b = 0 <=> a ⊥ b, a and b being non-zero vectors.

#### Important Points to be Remembered

(i) (a + b) * (a – b) = |a|^{2}2 – |b|^{2
}(ii) |a + b|^{2} = |a|^{2}2 + |b|^{2} + 2 (a * b)

(iii) |a – b|^{2} = |a|^{2}2 + |b|^{2} – 2 (a * b)

(iv) |a + b|^{2} + |a – b|^{2} = (|a|^{2}2 + |b|^{2}) and |a + b|^{2} – |a – b|^{2} = 4 (a * b)
or a * b = 1 / 4 [ |a + b|^{2} – |a – b|^{2} ]

(v) If |a + b| = |a| + |b|, then a is parallel to b.

(vi) If |a + b| = |a| – |b|, then a is parallel to b.

(vii) (a * b)^{2} ≤ |a|^{2}2 |b|^{2}

(viii) If a = a_{1}i + a_{2}j + a_{3}k, then |a|^{2} = a * a = a1^{2} + a2
^{ 2} + a3
^{ 2 }Or |a| = √a1 ^{2} + a2
^{2} + a3
^{2
}(ix) Angle between Two Vectors

If θ is angle between two non-zero vectors, a, b, then we
have
a * b = |a| |b| cos θ cos θ = a * b / |a| |b| If a = a_{1}i + a_{2}j + a_{3}k and b = b_{1}i + b_{2}j + b_{3}k

Then, the angle θ between a and b is given by cos θ = a * b / |a| |b| = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} / √a1
^{2} + a2
^{2} + a3
^{2} √b1
^{2} + b2
^{2} + b3
^{2
}(x) Projection and Component of a Vector

Projection of a on b = a * b / |a|

Projection of b on a = a * b / |a|

Vector component of a vector a on b

Similarly, the vector component of b on a = ((a * b) / |a^{2}|) * a

##### (xi) Work done by a Force

The work done by a force is a scalar quantity equal to the product of the magnitude of the force
and the resolved part of the displacement.

∴ F * S = dot products of force and displacement.

Suppose F_{1}, F_{1},…, F_{n} are n forces acted on a particle, then during the displacement S of the particle, the separate forces do quantities of work F_{1} * S, F_{2} * S, F_{n} * S.

Here, system of forces were replaced by its resultant R.

###### Vector or Cross Product of Two Vectors

The vector product of the vectors a and b is denoted by a * b and it is defined as

a * b = (|a| |b| sin θ) n = ab sin θ n …..(i)

where, a = |a|, b= |b|, θ is the angle between the vectors a and b and n is a unit vector which is
perpendicular to both a and b, such that a, b and n form a right-handed triad of vectors.

###### Important Points to be Remembered

(i) Let a = a_{1}i + a_{2}j + a_{3}k and b = b_{1}i + b_{2}j + b_{3}k

(ii) If a = b or if a is parallel to b, then sin θ = 0 and so a * b = 0.

(iii) The direction of a * b is regarded positive, if the rotation from a to b appears to be anticlockwise.

(iv) a * b is perpendicular to the plane, which contains both a and b. Thus, the unit vector perpendicular to both a and b or to the plane containing is given by n = a * b / |a * b| = a * b /
ab sin θ

(v) Vector product of two parallel or collinear vectors is zero.

(vi) If a * b = 0, then a = O or b = 0 or a and b are parallel on collinear.

(vii) Vector Product of Two Perpendicular Vectors If θ = 900, then sin θ = 1, i.e. , a * b = (ab)n or |a * b| = |ab n| = ab

(viii) Vector Product of Two Unit Vectors

If a and b are unit vectors, then

a = |a| = 1, b = |b| = 1

∴ a * b = ab sin θ n = (sin theta;).n

(ix) Vector Product is not Commutative

The two vector products a * b and b * a are equal in
magnitude but opposite in direction.

i.e., b * a =- a * b ……..(i)

(x) The vector product of a vector a with itself is null vector, i. e., a * a= 0.

(xi) **Distributive Law** For any three vectors a, b, c a * (b + c) = (a * b) + (a * c)

(xii) Area of a Triangle and Parallelogram

(a) The vector area of a ΔABC is equal to 1 / 2 |AB * AC| or 1 / 2 |BC * BA| or 1 / 2 |CB *
CA|.

(b) The area of a ΔABC with vertices having PV’s a, b, c respectively, is 1 / 2 |a * b + b * c + c
* a|.

(c) The points whose PV’s are a, b, c are collinear, if and only if a * b + b * c + c * a

(d) The area of a parallelogram with adjacent sides a and b is |a * b|.

(e) The area of a Parallelogram with diagonals a and b is 1 / 2 |a * b|.

(f) The area of a quadrilateral ABCD is equal to 1 / 2 |AC * BD|.

(xiii) Vector Moment of a Force about a Point

The vector moment of torque M of a force F about the point O is the vector whose magnitude is
equal to the product of |F| and the perpendicular distance of the point O from the line of action
of F.

∴ M = r * F

where, r is the position vector of A referred to O.

(a) The moment of force F about O is independent of the choice of point A on the line of action
of F.

(b) If several forces are acting through the same point A, then the vector sum of the moments of the separate forces about a point O is equal to the moment of their resultant force about O.

(xiv) The Moment of a Force about a Line

Let F be a force acting at a point A, O be any point on the given line L and a be the unit vector
along the line, then moment of F about the line L is a scalar given by (OA x F) * a

(xv) Moment of a Couple

(a) Two equal and unlike parallel forces whose lines of action are different are said to constitute a couple.

(b) Let P and Q be any two points on the lines of action of the forces – F and F, respectively.

The moment of the couple = PQ x F

#### Scalar Triple Product

If a, b, c are three vectors, then (a * b) * c is called scalar triple product and is denoted by [a b
c].

∴ [a b c] = (a * b) * c

#### Geometrical Interpretation of Scalar Triple Product

The scalar triple product (a * b) * c represents the volume of a parallelepiped whose
coterminous edges are represented by a, b and c which form a right handed system of vectors.

Expression of the scalar triple product (a * b) * c in terms of components

a = a_{1}i + a_{1}j + a_{1}k, b = a_{2}i + a_{2}j + a_{2}k, c = a_{3}i + a_{3}j + a_{3}k is

#### Properties of Scalar Triple Products

1. The scalar triple product is independent of the positions of dot and cross i.e., (a * b) * c = a *
(b * c).

2. The scalar triple product of three vectors is unaltered so long as the cyclic order of the vectors remains unchanged.

i.e., (a * b) * c = (b * c) * a= (c * a) * b **or** [a b c] = [b c a] = [c a b]

.3. The scalar triple product changes in sign but not in magnitude, when the cyclic order is changed.

i.e., [a b c] = – [a c b] etc.

4. The scalar triple product vanishes, if any two of its vectors are equal.

i.e., [a a b] = 0, [a b a] = 0 and [b a a] = 0.

5. The scalar triple product vanishes, if any two of its vectors are parallel or collinear.

6. For any scalar x, [x a b c] = x [a b c]. Also, [x a yb zc] = xyz [a b c].

7. For any vectors a, b, c, d, [a + b c d] = [a c d] + [b c d]

8. [i j k] = 1

11. Three non-zero vectors a, b and c are coplanar, if and only if [a b c] = 0.

12. Four points A, B, C, D with position vectors a, b, c, d respectively are coplanar, if and only
if [AB AC AD] = 0.

i.e., if and only if [b — a c— a d— a] = 0.

13. Volume of parallelepiped with three coterminous edges a, b,c is | [a b c] |.

14. Volume of prism on a triangular base with three coterminous edges a,b,c is 1/2 | [a b c] |.

15. Volume of a tetrahedron with three coterminous edges a, b,c is 1 / 6 | [a b c] |.

16. If a, b, c and d are position vectors of vertices of a tetrahedron, then Volume = 1 / 6 [b — a c — a d — a].

#### Vector Triple Product

If a, b, c be any three vectors, then (a * b) * c and a * (b * c) are known as vector triple product.

∴ a * (b * c)= (a * c)b — (a * b) c and (a * b) * c = (a * c)b — (b * c) a

#### Important Properties

(i) The vector r = a * (b * c) is perpendicular to a and lies in the plane b and c.

(ii) a * (b * c) ≠ (a * b) * c, the cross product of vectors is not associative.

(iii) a * (b * c)= (a * b) * c, if and only if and only if (a * c)b — (a * b) c = (a * c)b — (b * c) a,
if and only if c = (b * c) / (a * b) * a Or if and only if vectors a and c are collinear.

##### Reciprocal System of Ve ctors

Let a, b and c be three non-coplanar vectors and let a’ = b * c / [a b c], b’ = c * a / [a b c], c’ = a * b / [a b c]

Then, a’, b’ and c’ are said to form a reciprocal system of a, b and c.

##### Properties of Reciprocal System

(i) a * a’ = b * b’= c * c’ = 1

(ii) a * b’= a * c’ = 0, b * a’ = b * c’ = 0, c * a’ = c * b’= 0

(iii) [a’, b’, c’] [a b c] = 1 ⇒ [a’ b’ c’] = 1 / [a b c]

(iv) a = b’ * c’ / [a’, b’, c’], b = c’ * a’ / [a’, b’, c’], c = a’ * b’ / [a’, b’, c’]

Thus, a, b, c is reciprocal to the system a’, b’ ,c’.

(v) The orthonormal vector triad i, j, k form self reciprocal system.

(vi) If a, b, c be a system of non-coplanar vectors and a’, b’, c’ be the reciprocal system of
vectors, then any vector r can be expressed as r = (r * a’ )a + (r * b’)b + (r * c’)c.

###### Linear Combination of Vectors

Let a, b, c,… be vectors and x, y, z, … be scalars, then the expression x a yb + z c + … is
called a linear combination of vectors a, b, c,….

##### Collinearity of Three Points

The necessary and sufficient condition that three points with PV’s b, c are collinear is that there
exist three scalars x, y, z not all zero such that xa + yb + zc ⇒ x + y + z = 0.

##### Coplanarity of Four Points

The necessary and sufficient condition that four points with PV’s a, b, c, d are coplanar, if there
exist scalar x, y, z, t not all zero, such that xa + yb + zc + td = 0 rArr; x + y + z + t = 0.

If r = xa + yb + zc…

Then, the vector r is said to be a linear combination of vectors a, b, c,….

##### Linearly Independent and Dependent System of Vectors

(i) The system of vectors a, b, c,… is said to be linearly dependent, if there exists a scalars x, y,
z, … not all zero, such that xa + yb + zc + … = 0.

(ii) The system of vectors a, b, c, … is said to be linearly independent, if xa + yb + zc + td = 0
rArr; x + y + z + t… = 0.

#### Important Points to be Remembered

(i) Two non-collinear vectors a and b are linearly independent.

(ii) Three non-coplanar vectors a, b and c are linearly independent.

(iii) More than three vectors are always linearly dependent.

#### Resolution of Components of a Vector in a Plane

Let a and b be any two non-collinear vectors, then any vector r coplanar with a and b, can be
uniquely expressed as r = x a + y b, where x, y are scalars and x a, y b are called components of
vectors in the directions of a and b, respectively.

∴ Position vector of P(x, y) = x i + y j.

OP^{2} = OA^{2} + AP^{2} = |x|^{2} + |y|^{2} = x^{2} + y^{2}

OP = √x^{2} + y^{2}. This is the magnitude of OP.

where, x i and y j are also called resolved parts of OP in the directions of i and j, respectively.

##### Vector Equation of Line and Plane

(i) Vector equation of the straight line passing through origin and parallel to b is given by r = t
b, where t is scalar.

(ii) Vector equation of the straight line passing through a and parallel to b is given by r = a + t
b, where t is scalar.

(iii) Vector equation of the straight line passing through a and b is given by r = a + t(b – a),
where t is scalar.

(iv) Vector equation of the plane through origin and parallel to b and c is given by r = s b + t c,
where s and t are scalars.

(v) Vector equation of the plane passing through a and parallel to b and c is given by r = a + sb
+ t c, where s and t are scalars.

(vi) Vector equation of the plane passing through a, b and c is r = (1 – s – t)a + sb + tc, where s
and t are scalars.

##### Bisectors of the Angle between Two Lines

(i) The bisectors of the angle between the lines r = λa and r = μb are given by r = &lamba; (a /
|a| &plumsn; b / |b|)

(ii) The bisectors of the angle between the lines r = a + λb and r = a + μc are given by r = a + &lamba; (b / |b| &plumsn; c / |c|).