 # Direction of a Vector Formula

As the name suggests, when two distinct points are directed from one place to another then it is denoted by a vector. It can also be seen as differences between velocity and speed. We get no clue about in which direction the object is moving. Therefore, we use this formula that will enable us to know in which direction the object is moving. In physics, the magnitude and direction are expressed as a vector. If we say that the rock is moving at 5 meters per second, and the direction is towards the West, then it is represented using a vector.
If x is the horizontal movement and y is the vertical movement, then the formula of direction is:

$\LARGE \theta =\tan^{-1}\frac{y}{x}$

If (

$$\begin{array}{l}x_{1}\end{array}$$
,
$$\begin{array}{l}y_{1}\end{array}$$
) is the starting point and the ending point is (
$$\begin{array}{l}x_{2}\end{array}$$
,
$$\begin{array}{l}y_{2}\end{array}$$
), then the formula for direction is:

$$\begin{array}{l}\LARGE \theta =\tan^{-1}\frac{(y_{2}-y_{1})}{(x_{2}-x_{1})}\end{array}$$

### Solved Example

Question:

Find the direction of the vector

$$\begin{array}{l}\overrightarrow{PQ}\end{array}$$
whose initial point P is at (5, 2) and the endpoint is at Q is at (4, 3).

Solution:

Given

$$\begin{array}{l}(x_{1}\end{array}$$
,
$$\begin{array}{l}y_{1})\end{array}$$
=  (5, 2)

$$\begin{array}{l}(x_{2}\end{array}$$
,
$$\begin{array}{l}y_{2})\end{array}$$
(4, 3)

According to the formula we have,

$$\begin{array}{l}\theta\end{array}$$
=
$$\begin{array}{l}tan^{-1}\end{array}$$
$$\begin{array}{l}\frac{(y_{2} – y_{1})}{(x_{2} – x_{1})}\end{array}$$

$$\begin{array}{l}\theta\end{array}$$
=
$$\begin{array}{l}tan^{-1}\end{array}$$
$$\begin{array}{l}\frac{(3-2)}{(4-5)}\end{array}$$

θ = tan-1(-1)

θ = -45° or θ = 135°