Direction of Vector:

To fully describe the certain physical quantities, it is crucial to define both their magnitude and
direction. A vector is an entity, which comprises of both direction and magnitude. When two
separate points are directed from one location to another, they are represented by a vector. To
define a vector accurately, it's necessary to specify both its magnitude and direction."

Calculation of magnitude and direction of the Vector using two points:

The magnitude of a vector, from initial location/point A(x1, y1) to end location/point B(x2, y2),
can be calculated using following formula

√ (1)

The direction of the vector is specified using the angle, which is formed by vector with respect
to horizontal axis. The angle is formed by the vector, is calculated using the following formula

(2)

Example 1: Calculate the magnitude and direction of the vector from initial point (2,3) to end
point (5,4).

Solution: The direction can be calculated using equation (1).

The direction of the vector can be calculated using equation (2).

Example 2: Calculate the magnitude and direction of the vector from initial point (1,-1) to
endpoint (-2,-2).

Solution: The direction can be calculated using equation (1).

The direction of the vector can be calculated using equation (2).

It can be seen from calculation of example 1 and 2; both vectors have same magnitude and
direction. But, from graph, the direction of vector in example 2 is exactly opposite to the
direction of vector in example 2. Therefore, it is crucial to find the way to calculate actual angle
or actual direction of the vector. Following table should be followed to achieve actual angle
using the information of the .

Quadrant of Actual Angle

1
2
3
4

So the actual angle in example 2 is

Actual angle is
Practice Question:

Calculate the magnitudes and directions of the vectors pointed to

a. (3, 4) with respect to the origin.

b. (-3, 4) with respect to the origin.
c. (-3, -4) with respect to the origin and
d. (3, -4) with respect to the origin.

Solution (a): Here initial point is origin (0, 0) and endpoint is (3, 4), the
lies in the first quadrant.

The direction can be calculated using equation (1).

√ =5

The direction of the vector can be calculated using equation (2).

Solution (b): Here initial point is origin (0, 0) and endpoint is (-3, 4), the
lies in the second quadrant.

The direction can be calculated using equation (1).

√
√

√ =5

The direction of the vector can be calculated using equation (2).

As, the lies in the second quadrant, the actual angle will be

Actual angle will be,

Solution (c): Here initial point is origin (0, 0) and endpoint is (-3, -4), the
lies in the third quadrant.

The direction can be calculated using equation (1).

√ =5

The direction of the vector can be calculated using equation (2).

As, the lies in the third quadrant, the actual angle will be
Actual angle will be,

Solution (d): Here initial point is origin (0, 0) and endpoint is (3, -4), the
lies in the third quadrant.

The direction can be calculated using equation (1).

√ =5

The direction of the vector can be calculated using equation (2).

As, the lies in the third quadrant, the actual angle will be

Actual angle will be,