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Course: Precalculus > Unit 6

Lesson 7: Vector components from magnitude and direction
Math
Precalculus
Vectors
Vector components from magnitude and direction
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Converting between vector components and magnitude & direction review

Google Classroom
Review how to find a vector's magnitude and direction from its components and vice versa.

Cheat sheet

Vector magnitude from components

The magnitude of (a,b) is ||(a,b)||=a2+b2.

Vector direction from components

The direction angle of (a,b) is θ=tan−1⁡(ba) plus a correction based on the quadrant, according to this table:
QuadrantHow to adjust
Q1tan−1⁡(ba)
Q2tan−1⁡(ba)+180°
Q3tan−1⁡(ba)+180°
Q4tan−1⁡(ba)+360°

Vector components from magnitude & direction

The components of a vector with magnitude ||u→|| and direction θ are (||u→||cos⁡(θ),||u→||sin⁡(θ)).

What are vector magnitude and direction?

We are used to describing vectors in component form. For example, (3,4). We can plot vectors in the coordinate plane by drawing a directed line segment from the origin to the point that corresponds to the vector's components:
Considered graphically, there's another way to uniquely describe vectors — their magnitude and direction:
The magnitude of a vector gives the length of the line segment, while the direction gives the angle the line forms with the positive x-axis.
The magnitude of vector v→ is usually written as ||v→||.
Want to learn more about vector magnitude? Check out this video.
Want to learn more about vector direction? Check out this video.

Practice set 1: Magnitude from components

To find the magnitude of a vector from its components, we take the square root of the sum of the components' squares (this is a direct result of the Pythagorean theorem):
||(a,b)||=a2+b2
For example, the magnitude of (3,4) is 32+42=25=5.
Problem 1.1
u→=(1,7)
||u→||=

Either enter an expression with a square root symbol or a decimal rounded to the nearest hundredth.

Want to try more problems like this? Check out this exercise.

Practice set 2: Direction from components

To find the direction of a vector from its components, we take the inverse tangent of the ratio of the components:
θ=tan−1⁡(ba)
This results from using trigonometry in the right triangle formed by the vector and the x-axis.

Example 1: Quadrant I

Let's find the direction of (3,4):
tan−1⁡(43)≈53∘

Example 2: Quadrant IV

Let's find the direction of (3,−4):
tan−1⁡(−43)≈−53∘
The calculator returned a negative angle, but it's common to use positive values for the direction of a vector, so we must add 360∘:
−53∘+360∘=307∘

Example 3: Quadrant II

Let's find the direction of (−3,4). First, notice that (−3,4) is in Quadrant II.
tan−1⁡(4−3)≈−53∘
−53∘ is in Quadrant IV, not II. We must add 180∘ to obtain the opposite angle:
−53∘+180∘=127∘
Problem 2.1
u→=(5,8)
θ=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
∘
Enter your answer as an angle in degrees between 0∘ and 360∘ rounded to the nearest hundredth.

Want to try more problems like this? Check out this exercise.

Practice set 3: Components from magnitude and direction

To find the components of a vector from its magnitude and direction, we multiply the magnitude by the sine or cosine of the angle:
u→=(||u→||cos⁡(θ),||u→||sin⁡(θ))
This results from using trigonometry in the right triangle formed by the vector and the x-axis.
For example, this is the component form of the vector with magnitude 2 and angle 30∘:
(2cos⁡(30∘),2sin⁡(30∘))=(3,1)
Problem 3.1
u→≈( 
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
 ,
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
)
Round your final answers to the nearest hundredth.

Want to try more problems like this? Check out this exercise.

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