# NCERT Class 12-Mathematics: Chapter – 10 Vector Algebra Part 1 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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 and having 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