A vector can be formed by joining two distinct points. The point where the vector begins is called its tail, and the point where it ends is called its head.
- The direction of a vector is the angle it makes with the positive x-axis, measured counterclockwise from the tail.
- Direction is denoted by θ (theta).
Direction of a Vector Formula
The direction 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.
For a vector with horizontal displacement x and vertical displacement y:
θ = tan^{-1}(\frac{y}{x})
where,
- θ is the Direction of the Vector
- y is Vertical Displacement
- x is Horizontal Displacement
Direction between two points: 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))
Direction of a Vector Solved Examples
Example 1: Calculate the direction of the vector if the vertical displacement is 5 and the 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 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 6: 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 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°
Example 8: What is the direction of a vector A (3, 2) and B (5, 6) with respect to its initial point.
Solution:
Coordinates of Vector AB is given as x = (5-3) and y = (6-2)
Hence, tan θ = (y/x) = (4/2) = 2
θ = tan-1(2) = 63.435
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Practice Questions on Directions of a Vector
Q1. Find the direction of the vector if vertical displacement is 9 and horizontal displacement is 12.
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 (12, 3) and final point (5, 9).
Q4. Find the direction of the vector for the initial point (2, -3) and final point (-11, 0).