## 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 . Here, the point A from where the vector 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 or is called magnitude of or and it is represented by || or || or a.

Note: Since, the length is never negative, so the notation ||< 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 or is called the position vector of the point P with respect to O. The magnitude of or 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 Note: l^{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 . Note: Zero vector cannot be assigned a definite direction or it may be regarded as having any direction. The vectors , represent the zero vector.

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

**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. and are collinear, when or

**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 = , then 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 is negative of the vector and written as = – .

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, = Position vector of – Positron vector of

= – = –

**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 + = Note: The vector sum of three sides of a triangle taken in order is .

**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 and respectively, then we get

+ =

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

**Properties of vector addition**

**Commutative:** For vectors and , we have

**Associative:** For vectors , and , we have

Note: The associative property of vector addition enables us to write the sum of three vectors , and as without using brackets.

**Additive identity:** For any vector , a zero vector is its additive identity as

**Additive inverse:** For a vector , a negative vector of is its additive inverse as

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

Note: For any scalar λ, λ . = .

**Properties of Scalar Multiplication:** For vectors , and scalars p, q, we have

(i) p( + ) = p + p

(ii) (p + q) = p + q

(iii) p(q ) = (pq)

Note: To prove is parallel to , we need to show that = λ , where λ is a scalar.

**Components of a Vector:** Let the position vector of P with reference to O is , this form of any vector is-called its component form. Here, x, y and z are called the scalar components of and , and are called the vector components of along the respective axes.

**Two dimensions:** If a point P in a plane has coordinates (x, y), then , where and are unit vectors along OX and OY-axes, respectively.

Then,

**Three dimensions:** If a point P in a plane has coordinates (x, y, z), then , where , and are unit vectors along OX, OY and OZ-axes, respectively. 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 of point R, which divides the line segment joining the points A and B with position vectors and 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() and B() is given by

**Dot Product of Two Vectors:** If θ is the angle between two vectors and , then the scalar or dot product denoted by . is given by , where 0 ≤ θ ≤ π.

Note:

(i) is a real number

(ii) If either or , then θ is not defined.

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

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

where, is a unit vector perpendicular to both and , such that , and form a right handed system.

Note

(i) is a vector quantity, whose magnitude is

(ii) If either or , then0is not defined.

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