## CBSE 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 or simply as

**Magnitude of a Vector:** The length of the vector

Note: Since, the length is never negative, so the notation |

**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

**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 ^{2} + m^{2} + n^{2} = 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

**Unit vector:** A vector of unit length is called unit vector. The unit vector in the direction of

**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.

**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

**Negative vector:** Vector having the same magnitude but opposite in direction of the given vector, is called the negative vector e.g. Vector

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,

=

**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

**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

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

**Properties of vector addition**

**Commutative:** For vectors

**Associative:** For vectors

Note: The associative property of vector addition enables us to write the sum of three vectors

**Additive identity:** For any vector

**Additive inverse:** For a vector

**Multiplication of a Vector by a Scalar:** Let

Note: For any scalar λ, λ .

**Properties of Scalar Multiplication:** For vectors

(i) p(

(ii) (p + q)

(iii) p(q

Note: To prove

**Components of a Vector:** Let the position vector of P with reference to O is

**Two dimensions:** If a point P in a plane has coordinates (x, y), then

Then,

**Three dimensions:** If a point P in a plane has coordinates (x, y, z), then

**Vector Joining of Two Points:** If 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

**Section Formula:** Position vector

For external division,

Note: Position vector of mid-point of the line segment joining end points A(

**Dot Product of Two Vectors:** If θ is the angle between two vectors

Note:

(i)

(ii) If either

Properties of dot product of two vectors

**Vector (or Cross) Product of Vectors:** If θ is the angle between two non-zero, non-parallel vectors

where,

Note

(i)

(ii) If either

Properties of cross product of two vectors