NCERT Class 12-Mathematics: Chapter –10 Vector Algebra Part 1

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10.1 Overview

10.1.1 A quantity that has magnitude as well as direction is called a vector.

10.1.2 The unit vector in the direction of is given by and is represented .

10.1.3 Position vector of a point is given as and its magnitude as , where is the origin.

10.1.4 The scalar components of a vector are its direction ratios, and represent its projections along the respective axes.

10.1.5 The magnitude r, direction ratios and direction cosines of any vector are related as:

10.1.6 The sum of the vectors representing the three sides of a triangle taken in order is

10.1.7 The triangle law of vector addition states that “If two vectors are represented by two sides of a triangle taken in order, then their sum or resultant is given by the third side taken in opposite order”.

10.1.8 Scalar multiplication

If is a given vector and a scalar, then is a vector whose magnitude is . The direction of is same as that of if is positive and, opposite to that of if is negative.

10.1.9 Vector joining two points

If and are any two points, then

10.1.10 Section formula

The position vector of a point R dividing the line segment joining the points P and Q whose position vectors are and

(i) In the ratio internally, is given by

(ii) in the ratio externally, is given by

10.1.11 Projection of along is and the Projection vector of along is

10.1.12 Scalar or dot product

The scalar or dot product of two given vectors andhaving an angle θ between them is defined as

10.1.13 Vector or cross product

The cross product of two vectors and having angle θ between them is given as

where is a unit vector perpendicular to the plane containing and and , form a right handed system.

10.1.14 and are two vectors and is any scalar, then

Angle between two vectors and is given by

10.2 Solved Examples

Short Answer (S.A)

Question 1:

Find the unit vector in the direction of the sum of the vectors

Answer:

Let denote the sum of and We have

Now .

Thus, the required unit vector is

Question 2:

Find a vector of magnitude 11 in the direction opposite to that of where P and Q are the points and , respectively.

Thus

Therefore, unit vector in the direction of is given by

Hence, the required vector of magnitude 11 in direction of is