**Summary**

- Unit vector is defined as:
- Position vector of two points is:
- Equation of a vector line is where
*a*is the point on the line and*b*is the direction - Angle between two vectors is

Vector quantities are extremely useful in physics. The important characteristic of a vector quantity is that it has both a magnitude and a direction. Both of these properties must be given in order to specify a vector completely.

#### Scaler & Vector

A scalar quantity is the one which is completely represented by its magnitude, in terms of a given unit i.e mass (kg), energy (joules) etc are examples of a scalar quantity.

Whereas a vector quantity is the one which is completely represented by its magnitude in terms of a given unit and its direction from point of reference i.e acceleration, displacement, velocity etc.

Let *r* be a given vector. Using the above definition of a vector quantity, we can write vector *r* as:

Where is the magnitude of the vector *r* and is the unit vector in the direction of *r*.

#### Unit vector

A unit vector is the vector whose magnitude is 1 unit. It is used to specify the direction of the given vector. Also:

i.e

If a vector is divided by its magnitude (modulus) then we get a unit vector in the direction of that vector.

Unit vectors can be described as *i + j*, where* i* is the direction of the *x axis* and* j* is the direction of the* y axis*.

If *r = p i +q j*, then the unit vector in the direction of *r* is given by:

#### Position Vector

If ”O” is the origin then the position of any point A can be determined by the vector which is called the position vector of A with respect to the origin O.

We can say that if *a* is the position vector of the point A, then:

Let *a* and* b* be two position vectors of the points A and B with respect to the origin O respectively as shown in Fig 1.

Then:

and

Therefore:

Similarly, we can express other vectors in terms of their position vectors i.e:

,

, etc

#### Magnitude of a vector

For example for a given vector* a i + b j*, we can find its magnitude using the formula:

#### Example #1

Q. Find the magnitude of the vector *5 i + 4 j*

*Solution:*

** Ans**

#### Vector Equation of a line

Let’s suppose a line is parallel to a direction vector *b* and passes through fixed point A with position vector *a*. Let* r* be the position vector of a general point on the line.

Thus the equation of the line is defined by where is a parameter and corresponds to a point on the line.

#### Example #2

Q. Find the equation of the line through the points (2, -3) in the direction *3 i – 5 j*

*Solution:*

We know the point* a = (2,-3) *and *b = (3, -5)*

Substitute value in the above equation we get:

#### Scalar Product

When the product of two vectors is scalar, it is called a scalar product or dot product.

Remember these points about the scalar product;

1. Scalar product is commutative:

2. Scalar product is distributive: where *a, b, c* are three vectors

Suppose we are given two vectors:

Then the scalar product of *p* and *q* would be:

#### Angle between two vectors

If is the angle between two vectors a and b, then:

or can be written as

*Note: When the dot product is equal to zero, it means two vectors are perpendicular to each other as cos 90°=0*