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) = \((x_2x_1,y_2y_1)\)
 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) = \((x_2x_1,y_2y_1)\) = (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°.