The direction of a vector is the angle made by the vector with the horizontal axis, that is, the X-axis. 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 |
The direction of a vector is the orientation of the vector, that is, the angle it makes with the x-axis. 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 \(\overrightarrow = |a|\hat\), where |a| denotes the magnitude of the vector, whereas \(\hat\) is a unit vector and denotes the direction of the vector a.
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):
| 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 (x1, y1) and (x2, y2), then to find its direction:
Let us now go through some examples to understand 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) = (3-1, 9-4) = (2, 5). The direction of the vector is given by the formula,
= 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
Related Topics on Direction of a Vector
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 (x1, y1) = (1, 3). (x2, y2) = (-4, -2). The vector is, (x, y) = (x2 - x1, y2 - y1) = (-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°
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