# 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