# Direction of a Vector Formula

Before we learn the direction of a vector formula, let us recall what is a vector. A vector is a physical quantity having direction and magnitude both. The magnitude of a vector is its length whereas the direction of a vector is the angle made by it with the horizontal. Let us learn the direction of a vector formula along with a few solved examples.

## What Is the Direction of a Vector Formula?

The direction of a vector formula is related to the slope of a line. We know that the slope of a line that passes through the origin and a point (x, y) is y/x. We also know that if θ is the angle made by this line, then its slope is tan θ, i.e., tan θ = y/x. Hence, θ = tan^{-1} (y/x). Thus, the direction of a vector (x, y) is found using the formula tan^{-1} (y/x) but while calculating this angle, the quadrant in which (x, y) lies also should be considered. So we find the direction of a vector using the following steps.

To find the direction of a vector (x, y):

- Find α using α = tan
^{-1}|y/x|. - Find the direction of the vector θ using the following rules depending on which quadrant (x, y) lies in:
Quadrant in which (x, y) lies θ (in degrees) 1 α 2 180 - α 3 180 + α 4 360 - α

To find the direction of a vector whose endpoints are given by the position vectors \((x_1,y_1)\) and \((x_2,y_2)\), then to find its direction:

- Find (x, y) using (x, y) = \((x_2-x_1,y_2-y_1)\)
- Find α and θ just as explained earlier.

Let us see the applications of the direction of a vector formula in the following solved examples.

## Solved Examples Using Direction of a Vector Formula

**Example 1:** Find the direction of the vector (1, -√3) using the direction of a vector formula.

**Solution:**

Given (x, y) = (1, -√3).

We first find α using α = tan^{-1} |y/x|.

α = tan^{-1} |-√3/1| = tan^{-1} √3 = 60°.

We know that (1, -√3) lies in quadrant 4. Thus, the direction of the given vector is,

θ = 360 - α = 360 - 60 = 300°.

**Answer: **The direction of the given vector = 300°.

**Example 2:** Find the direction of the vector which starts at (1, 3) and ends at (-4, -2).

**Solution:**

Given

\((x_1,y_1)\) = (1, 3).

\((x_2,y_2)\) = (-4, -2).

The vector is, (x, y) = \((x_2-x_1,y_2-y_1)\) = (-4 - 1, -2 - 3) = (-5, -5).

By using the direction of a vector formula,

α = tan^{-1} |-5/-5| = tan^{-1} 1 = 45°.

We know that (-5, -5) lies in quadrant 3. Thus, the direction of the given vector is,

θ = 180 + α = 180 + 45 = 225°.

**Answer: **The direction of the given vector = 225°.