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## Get ready for AP® Calculus

### Unit 8: Lesson 7

Vector components from magnitude & direction

# Converting between vector components and magnitude & direction review

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 left parenthesis, a, comma, b, right parenthesis is vertical bar, vertical bar, left parenthesis, a, comma, b, right parenthesis, vertical bar, vertical bar, equals, square root of, a, squared, plus, b, squared, end square root.

### Vector direction from components

The direction angle of left parenthesis, a, comma, b, right parenthesis is theta, equals, tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, b, divided by, a, end fraction, right parenthesis plus a correction based on the quadrant, according to this table:
Q1tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, b, divided by, a, end fraction, right parenthesis
Q2tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, b, divided by, a, end fraction, right parenthesis, plus, 180, degree
Q3tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, b, divided by, a, end fraction, right parenthesis, plus, 180, degree
Q4tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, b, divided by, a, end fraction, right parenthesis, plus, 360, degree

### Vector components from magnitude & direction

The components of a vector with magnitude vertical bar, vertical bar, u, with, vector, on top, vertical bar, vertical bar and direction theta are left parenthesis, vertical bar, vertical bar, u, with, vector, on top, vertical bar, vertical bar, cosine, left parenthesis, theta, right parenthesis, comma, vertical bar, vertical bar, u, with, vector, on top, vertical bar, vertical bar, sine, left parenthesis, theta, right parenthesis, right parenthesis.

## What are vector magnitude and direction?

We are used to describing vectors in component form. For example, left parenthesis, 3, comma, 4, right parenthesis. 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 start color #11accd, start text, m, a, g, n, i, t, u, d, e, end text, end color #11accd and start color #1fab54, start text, d, i, r, e, c, t, i, o, n, end text, end color #1fab54:
The start color #11accd, start text, m, a, g, n, i, t, u, d, e, end text, end color #11accd of a vector gives the length of the line segment, while the start color #1fab54, start text, d, i, r, e, c, t, i, o, n, end text, end color #1fab54 gives the angle the line forms with the positive x-axis.
The magnitude of vector v, with, vector, on top is usually written as vertical bar, vertical bar, v, with, vector, on top, vertical bar, vertical bar.

## 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):
vertical bar, vertical bar, left parenthesis, a, comma, b, right parenthesis, vertical bar, vertical bar, equals, square root of, a, squared, plus, b, squared, end square root
For example, the magnitude of left parenthesis, 3, comma, 4, right parenthesis is square root of, 3, squared, plus, 4, squared, end square root, equals, square root of, 25, end square root, equals, 5.
Problem 1.1
u, with, vector, on top, equals, left parenthesis, 1, comma, 7, right parenthesis
vertical bar, vertical bar, u, with, vector, on top, vertical bar, vertical bar, equals

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:
theta, equals, tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, b, divided by, a, end fraction, right parenthesis
This results from using trigonometry in the right triangle formed by the vector and the x-axis.

### Example 1: Quadrant $\text{I}$start text, I, end text

Let's find the direction of left parenthesis, 3, comma, 4, right parenthesis:
tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, 4, divided by, 3, end fraction, right parenthesis, approximately equals, 53, degrees

### Example 2: Quadrant $\text{IV}$start text, I, V, end text

Let's find the direction of left parenthesis, 3, comma, minus, 4, right parenthesis:
tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, minus, 4, divided by, 3, end fraction, right parenthesis, approximately equals, minus, 53, degrees
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, degrees:
minus, 53, degrees, plus, 360, degrees, equals, 307, degrees

### Example 3: Quadrant $\text{II}$start text, I, I, end text

Let's find the direction of left parenthesis, minus, 3, comma, 4, right parenthesis. First, notice that left parenthesis, minus, 3, comma, 4, right parenthesis is in Quadrant start text, I, I, end text.
tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, 4, divided by, minus, 3, end fraction, right parenthesis, approximately equals, minus, 53, degrees
minus, 53, degrees is in Quadrant start text, I, V, end text, not start text, I, I, end text. We must add 180, degrees to obtain the opposite angle:
minus, 53, degrees, plus, 180, degrees, equals, 127, degrees
Problem 2.1
u, with, vector, on top, equals, left parenthesis, 5, comma, 8, right parenthesis
theta, equals
degrees
Enter your answer as an angle in degrees between 0, degrees and 360, degrees 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, with, vector, on top, equals, left parenthesis, vertical bar, vertical bar, u, with, vector, on top, vertical bar, vertical bar, cosine, left parenthesis, theta, right parenthesis, comma, vertical bar, vertical bar, u, with, vector, on top, vertical bar, vertical bar, sine, left parenthesis, theta, right parenthesis, right parenthesis
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 start color #11accd, 2, end color #11accd and angle start color #1fab54, 30, degrees, end color #1fab54:
left parenthesis, start color #11accd, 2, end color #11accd, cosine, left parenthesis, start color #1fab54, 30, degrees, end color #1fab54, right parenthesis, comma, start color #11accd, 2, end color #11accd, sine, left parenthesis, start color #1fab54, 30, degrees, end color #1fab54, right parenthesis, right parenthesis, equals, left parenthesis, square root of, 3, end square root, comma, 1, right parenthesis
Problem 3.1
u, with, vector, on top, approximately equals, left parenthesis, space
space, comma
right parenthesis