Direction of a Vector
The direction of a vector is the angle made by the vector with the horizontal axis, that is, the Xaxis. The direction of a vector is given by the counterclockwise rotation of the angle of the vector about its tail due east. For example, a vector with a direction of 45 degrees is a vector that has been rotated 45 degrees in a counterclockwise direction relative to due east. Another convention to express the direction of a vector is as an angle of rotation of the vector about its tail from east, west, north or south. For example, if the direction of a vector is 60 degrees North of West, it implies that the vector pointing West has been rotated 60 degrees towards the northern direction.
The direction along which a vector act is defined as the direction of a vector. Let us learn the direction of a vector formula and how to determine the direction of a vector in different quadrants along with a few solved examples.
1.  What is the Direction of a Vector? 
2.  Direction of a Vector Formula 
3.  How to Find the Direction of a Vector? 
4.  FAQs on Direction of a Vector 
What is the Direction of a Vector?
The direction of a vector is the orientation of the vector, that is, the angle it makes with the xaxis. A vector is drawn by a line with an arrow on the top and a fixed point at the other end. The direction in which the arrowhead of the vector is directed gives the direction of the vector. For example, velocity is a vector. It gives the magnitude at which the object is moving along with the direction towards which the object is moving. Similarly, the direction in which a force is applied is given by the force vector. The direction of a vector is denoted by \(\overrightarrow{a} = a\hat{a}\), where a denotes the magnitude of the vector, whereas \(\hat{a}\) is a unit vector and denotes the direction of the vector a.
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.
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 the vector (x, y) using the formula (x, y) = (x_{2}  x_{1}, y_{2}  y_{1})
 Find α and θ just as explained earlier.
Let us now go through some examples to understand how to find the direction of a vector.
How to Find the Direction of a Vector?
Now that we know the formulas to determine the direction of a vector in different quadrants, let us go through an example to understand the application of the formula.
Example 1: Determine the direction of the vector with initial point P = (1, 4) and Q = (3, 9).
To determine the direction of the vector PQ, let us first determine the coordinates of the vector PQ
(x, y) = (31, 94) = (2, 5). The direction of the vector is given by the formula,
θ = tan^{1} 5/2
= 68.2° [Because (2, 5) lies in the first quadrant]
The direction of the vector is given by 68.2°.
Example 2: Consider the image given below.
The vector in the above image makes an angle of 50° in the counterclockwise direction with the east. Hence, the direction of the vector is 50° from the east.
Important Notes on Direction of a Vector
 The direction of a vector can be expressed by the angle its tail forms with east, north, west or south.
 After determining the value of tan^{1} y/x, we can apply the respective formula for each quadrant.
 The direction of a vector is can also be given by the angle made by the vector in the counterclockwise direction from east.
Related Topics on Direction of a Vector
Direction of a Vector Examples

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°
FAQs on Direction of a Vector
What is the Direction of a Vector?
The direction of a vector is the angle made by the vector with the horizontal axis, that is, the Xaxis.
What is the Direction of a Vector Formula?
To find the direction of a vector (x, y):
 Find α using α = tan^{1} y/x
 The direction of the vector (x, y) is given by:
 α, if (x, y) lies in the first quadrant
 180°  α, if (x, y) lies in the second quadrant
 180° + α, if (x, y) lies in the third quadrant
 360°  α, if (x, y) lies in the fourth quadrant
How to Find the Direction of a Vector?
The direction of a vector can be calculated using the formulas for each quadrant.
What does the Direction of a Vector Represent?
The direction of a vector represents the direction towards which the object is moving.
How Do you Find the Direction of a Vector Given its Components?
The direction of a vector can be determined by checking the quadrant in which the vector lies and then applying the corresponding formula.
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