Chapter 10 Vector Algebra | class 12th | quick revision notes maths

Class 12 Maths Notes Chapter 10 Vector Algebra

Vector: Those quantities which have magnitude, as well as direction, are called vector quantities or vectors.
Note: Those quantities which have only magnitude and no direction, are called scalar quantities.

Representation of Vector: A directed line segment has magnitude as well as direction, so it is called vector denoted as AB→ or simply as a⃗ . Here, the point A from where the vector AB→ starts is called its initial point and the point B where it ends is called its terminal point.

Magnitude of a Vector: The length of the vector AB→ or a⃗  is called magnitude of AB→ or a⃗  and it is represented by |AB→| or |a⃗ | or a.
Note: Since, the length is never negative, so the notation |a⃗ |< 0 has no meaning.

Position Vector: Let O(0, 0, 0) be the origin and P be a point in space having coordinates (x, y, z) with respect to the origin O. Then, the vector OP→ or r⃗  is called the position vector of the point P with respect to O. The magnitude of OP→ or r⃗  is given by

Direction Cosines: If α, β and γ are the angles which a directed line segment OP makes with the positive directions of the coordinate axes OX, OY and OZ respectively, then cos α, cos β and cos γ are known as the direction cosines of OP and are generally denoted by the letters l, m and n respectively.

i.e. l = cos α, m = cos β, n = cos γ Let l, m and n be the direction cosines of a line and a, b and c be three numbers, such that la=mb=nc=r⃗  Note: l² + m² + n² = 1

Types of Vectors
Null vector or zero vector: A vector, whose initial and terminal points coincide and magnitude is zero, is called a null vector and denoted as 0⃗ . Note: Zero vector cannot be assigned a definite direction or it may be regarded as having any direction. The vectors AA→ , BB→ represent the zero vector.

Unit vector: A vector of unit length is called unit vector. The unit vector in the direction of a⃗  is a^=a⃗ ∣∣a⃗ ∣∣

Collinear vectors: Two or more vectors are said to be collinear, if they are parallel to the same line, irrespective of their magnitudes and directions, e.g. a⃗  and b⃗  are collinear, when a⃗ =±λb⃗  or |a|→=λ|b|→

Coinitial vectors: Two or more vectors having the same initial point are called coinitial vectors.

Equal vectors: Two vectors are said to be equal, if they have equal magnitudes and same direction regardless of the position of their initial points. Note: If a⃗  = b⃗ , then |a|→=|b|→ but converse may not be true.

Negative vector: Vector having the same magnitude but opposite in direction of the given vector, is called the negative vector e.g. Vector BA→ is negative of the vector AB→ and written as BA→ = – AB→.
Note: The vectors defined above are such that any of them may be subject to its parallel displacement without changing its magnitude and direction. Such vectors are called ‘free vectors’.

To Find a Vector when its Position Vectors of End Points are Given: Let a and b be the position vectors of end points A and B respectively of a line segment AB. Then, AB→ = Position vector of B⃗  – Positron vector of A⃗ 
= OB→ – OA→ = b⃗  – a⃗ 

Addition of Vectors
Triangle law of vector addition: If two vectors are represented along two sides of a triangle taken in order, then their resultant is represented by the third side taken in opposite direction, i.e. in ∆ABC, by triangle law of vector addition, we have BC→ + CA→ = BA→ Note: The vector sum of three sides of a triangle taken in order is 0⃗ .

Parallelogram law of vector addition: If two vectors are represented along the two adjacent sides of a parallelogram, then their resultant is represented by the diagonal of the sides. If the sides OA and OC of parallelogram OABC represent OA→ and OC→ respectively, then we get
OA→ + OC→ = OB→

Note: Both laws of vector addition are equivalent to each other.

Properties of vector addition
Commutative: For vectors a⃗  and b⃗ , we have a⃗ +b⃗ =b⃗ +a⃗ 

Associative: For vectors a⃗ , b⃗  and c⃗ , we have a⃗ +(b⃗ +c⃗ )=(a⃗ +b⃗ )+c⃗ 
Note: The associative property of vector addition enables us to write the sum of three vectors a⃗ , b⃗  and c⃗  as a⃗ +b⃗ +c⃗  without using brackets.

Additive identity: For any vector a⃗ , a zero vector 0⃗  is its additive identity as a⃗ +0⃗ =a⃗ 

Additive inverse: For a vector a⃗ , a negative vector of a⃗  is its additive inverse as a⃗ +(−a→)=0⃗ 

Multiplication of a Vector by a Scalar: Let a⃗  be a given vector and λ be a scalar, then multiplication of vector a⃗  by scalar λ, denoted as λ a⃗ , is also a vector, collinear to the vector a⃗  whose magnitude is |λ| times that of vector a⃗  and direction is same as a⃗ , if λ > 0, opposite of a⃗ , if λ < 0 and zero vector, if λ = 0.
Note: For any scalar λ, λ . 0⃗  = 0⃗ .

Properties of Scalar Multiplication: For vectors a⃗ , b⃗  and scalars p, q, we have
(i) p(a⃗  + a⃗ ) = p a⃗  + p a⃗ 
(ii) (p + q) a⃗  = p a⃗  + q a⃗ 
(iii) p(q a⃗ ) = (pq) a⃗ 
Note: To prove a⃗  is parallel to b⃗ , we need to show that a⃗  = λ a⃗ , where λ is a scalar.

Components of a Vector: Let the position vector of P with reference to O is OP→=r⃗ =xi^+yj^+zk^, this form of any vector is-called its component form. Here, x, y and z are called the scalar components of r⃗  and xi^, yj^ and zk^ are called the vector components of r⃗  along the respective axes.

Two dimensions: If a point P in a plane has coordinates (x, y), then OP→=xi^+yj^, where i^ and j^ are unit vectors along OX and OY-axes, respectively.
Then, ∣∣∣OP→∣∣∣=x2+y2−−−−−−√

Three dimensions: If a point P in a plane has coordinates (x, y, z), then OP→=xi^+yj^+zk^, where i^, j^ and k^ are unit vectors along OX, OY and OZ-axes, respectively. Then, ∣∣∣OP→∣∣∣=x2+y2+z2−−−−−−−−−−√

Vector Joining of Two Points: If P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) are any two points, then the vector joining P₁ and P₂ is the vector P1P2→

Section Formula: Position vector OR→ of point R, which divides the line segment joining the points A and B with position vectors a⃗  and b⃗  respectively, internally in the ratio m : n is given by

For external division,

Note: Position vector of mid-point of the line segment joining end points A(a⃗ ) and B(b⃗ ) is given by OR→=a⃗ +b⃗ 2

Dot Product of Two Vectors: If θ is the angle between two vectors a⃗  and b⃗ , then the scalar or dot product denoted by a⃗  . b⃗  is given by a⃗ ⋅b⃗ =|a⃗ |∣∣b⃗ ∣∣cosθ, where 0 ≤ θ ≤ π.
Note:
(i) a⃗ ⋅b⃗  is a real number
(ii) If either a⃗ =0⃗  or b⃗ =0⃗ , then θ is not defined.

Properties of dot product of two vectors a⃗  and b⃗  are as follows:

Vector (or Cross) Product of Vectors: If θ is the angle between two non-zero, non-parallel vectors a⃗  and b⃗ , then the cross product of vectors, denoted by a⃗ ×b⃗  is given by

where, n^ is a unit vector perpendicular to both a⃗  and b⃗ , such that a⃗ , b⃗  and n^ form a right handed system.
Note
(i) a⃗ ×b⃗  is a vector quantity, whose magnitude is ∣∣a⃗ ×b^∣∣=|a⃗ ||b|→sinθ
(ii) If either a⃗ =0⃗  or b⃗ =0⃗ , then0is not defined.

Properties of cross product of two vectors a⃗  and b⃗  are as follows: