Direction of a Vector

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.

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 \(\overrightarrow{a} = |a|\hat{a}\), where |a| denotes the magnitude of the vector, whereas \(\hat{a}\) is a unit vector and denotes the direction of the vector a.

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:

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:

• Find the vector (x, y) using the formula (x, y) = (x₂ - x₁, y₂ - y₁)
• 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.

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,

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

• Vectors
• Vector Formulas
• Cross Product

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 X-axis.

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.