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Direction of a Vector Formula

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A vector is formed when two distinct points are joined with each other. The point from which a line was drawn to the other point determines the direction of that vector. A vector’s direction is the angle formed by the vector with the horizontal axis, also known as the X-axis. It is provided by the counterclockwise rotation of the vector’s angle about its tail due east. In other words, the orientation of a vector, that is, the angle it makes with the x-axis, is defined as its direction.

In this article we will learn about, Direction of Vector Definition, Direction of Vector Formula, Magnitude of Vector, Examples, and others in detail.

What is the Direction of a Vector?

Direction of a vector is orientation of the vector in the 3-D space. Suppose we are given a vector from point A to B then the arrow in the direction from A to B shows the direction of the vector.

The direction of the vector is defined as the angle that the vector makes with the horizontal line. We denote the direction of the vector by applying the arrow over the vector in the direction from the first point to another.

Direction of a Vector Formula

The direction of a vector is denoted by the symbol θ. Its formula equals the inverse tangent of the ratio of the distance moved by the line along the y-axis to the distance moved along the x-axis. To put it another way, it is the inverse tangent of the slope of the line.

θ = tan-1(y/x)

where,

  • θ is the Direction of Vector
  • y is the Vertical Displacement
  • x is the Horizontal Displacement

For a vector line with starting point (x1, y1) and final point (x2, y2), the direction is given by,

θ = tan-1 ((y2 – y1) / (x2 – x1))

How to Find the Direction of a Vector?

Direction of a vector is easily find using the direction of the vector formula added above. Let’s understand with the help of an example

Example: What is the direction of a vector A (3, 2) and B (5, 6) with respect to its initial point.

Solution:

The coordinates of Vector AB is given as x = (5-3) and y = (6-2)

Hence, tan θ = (y/x) = (4/2) = 2

There θ = tan-1(2) = 63.435

What is Magnitude of a Vector?

Magnitude of a vector is the distance between the initial point and the final point of the vector. Suppose we are given a vector \vec{AB}  then the maginuted of the vector is the distance between point A and point B. The magnitude of the vector is written as, |\vec{AB}  |.

Read More,

Direction of a Vector Examples

Example 1: Calculate the direction of the vector if vertical displacement is 5 and horizontal displacement is 4. 

Solution:

We have,

  • y = 5
  • x = 4

Using the formula we get,

θ = tan-1 (y/x)

θ = tan-1 (5/4) = 51.34°

Example 2: Calculate the direction of the vector if vertical displacement is 7 and horizontal displacement is 5.

Solution:

We have,

  • y = 7
  • x = 5

Using the formula we get,

θ = tan-1 (y/x)

θ = tan-1 (7/5) = 54.46°

Example 3: Calculate the vertical displacement if the direction of the vector is 60° and the horizontal displacement is 5.

Solution:

We have,

  • θ = 60°
  • x = 5

Using the formula we get,

tan θ = y/x

=> y = x tan θ

y = 5 tan 60° = 8.66

Examples 4: Calculate the vertical displacement if the direction of the vector is 30° and the horizontal displacement is 8.

Solution:

We have,

  • θ = 30°
  • x = 8

Using the formula we get,

tan θ = y/x

=> y = x tan θ

y = 8 tan 30° = 4.61

Example 5: Calculate the horizontal displacement if the direction of the vector is 50° and the vertical displacement is 4.

Solution:

We have,

  • θ = 50°
  • x = 4

Using the formula we get,

tan θ = y/x

=> x = y/tan θ

x = 4/tan 50° = 3.35

Example 6: Calculate the horizontal displacement if the direction of the vector is 45° and the vertical displacement is 9.

Solution:

We have,

  • θ = 45°
  • x = 9

Using the formula we get,

tan θ = y/x

=> x = y/tan θ

x = 9/tan 45° = 9/1 = 9

Example 7: Calculate the direction of the vector for the initial point (8, 4) and final point (10, 6).

Solution:

We have,

  • (x1, y1) = (8, 4)
  • (x2, y2) = (10, 6)

Find the vertical displacement.

y = y2 – y1 

= 6 – 4 = 2

Find the horizontal displacement.

x = x2 – x1

x = 10 – 8 = 2

Using the formula we get,

θ = tan-1 (y/x)

θ = tan-1 (2/2)

θ = tan-1 (1) = 45°

Practice Questions on Directions of a Vector

Q1: Find the direction of the vector for the initial point (12, 3) and final point (5, 9).

Q2: Find the horizontal displacement if the direction of the vector is 60° and the vertical displacement is 12.

Q3: Find the direction of the vector for the initial point (2, -3) and final point (-11, 0).

Q4: Find the direction of the vector if vertical displacement is 9 and horizontal displacement is 12.

Direction of a Vector-FAQs

1. What is the Direction of a Vector?

The direction of a vector is defined as the angle made by the vector with the horizontal axis (i.e. X-axis).

2. What is the Direction of a Vector Formula?

The direction of the vector represented with the position vector (x, y) is easily found using the formula, tan-1(y/x).

3. How to Find the Direction of a Vector?

The direction of a vector is calculated using the direction of vector formula added above.

4. How do You Find the Direction of a Vector Product?

The direction of the vector product is found using the Right Hand Screw Rule.



Last Updated : 25 Nov, 2023
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