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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:

Quadrant | How to adjust |
---|---|

Q1 | tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, b, divided by, a, end fraction, right parenthesis |

Q2 | tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, b, divided by, a, end fraction, right parenthesis, plus, 180, degree |

Q3 | tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, b, divided by, a, end fraction, right parenthesis, plus, 180, degree |

Q4 | tangent, 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.

*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):

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.

*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:

This results from using trigonometry in the right triangle formed by the vector and the x-axis.

### Example 1: Quadrant start text, I, end text

Let's find the direction of left parenthesis, 3, comma, 4, right parenthesis:

### Example 2: Quadrant start text, I, V, end text

Let's find the direction of left parenthesis, 3, comma, minus, 4, right parenthesis:

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:

### Example 3: Quadrant 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.

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:

*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:

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:

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

## Want to join the conversation?

- problem2.2 has two answers,am i right?

325.01 & 145.01(1 vote)- I think there's only one answer that fits the question. You are right that arctan(7/-10) yields two answers in the range 0-360 degrees, but the vector u is in the
**second**quadrant (u=-10i+7j), and so its angle cannot be 325 degrees, even though a vector with that angle has the same slope/tangent value. A vector with a direction of 325 degrees would be in the fourth quadrant.(23 votes)

- What is the pedagogical reason for using parenthesis vector notation? I find that for a first introduction to vectors, it is hard for students to understand how vectors are different from ordered pairs. Using the identical notation, just makes the situation worse. Why not use matrix notation (x above y) or angle vector brackets?(4 votes)
- It may be to underline the fact that there really is no difference between a vector and a standard ordered pair. The only real difference is that we've defined addition between vectors.(5 votes)

- How do you calculate the component form of a vector in 3D (x,y,z)?(3 votes)
- If you have all three coordinates it is super simple. xi + jy + zk. If you only have the magnitude and some angle, you would actually need two angles, usually from two of the three planes. the xy plane, xz plane or yz plane.(3 votes)

- What about 3 Dimensional Vectors? How would I calculate and express the angle it builds with the planes of the origin?(3 votes)
- How do I find which quadrant the vector would be in? I think i missed something along the way.(2 votes)
- By Checking the Value of "a" and "b" in the Vector we can Understand the Quadrant of the Vector...Since the Values of (a,b) are Co-ordinates in the Co-ordinate Plane(3 votes)

- If vector A = 12i – 16j and vector B = –24i + 10j, what is the direction of the vector C = 2A – B?(2 votes)
- Vector addition, or subtraction, is just combining steps in the various directions. then finding the direction is taking the inverse tangent of the ratio of the combined j steps over the combined i steps.

Your question says 2A - B. if this were just A - B you would do 12i - 16j -(-24i + 10i). Here A is multiplied by 2, so let's get that done first.

2A

2(12i-16j)

24i - 32j

There, now we can do the subtraction part

2A - B

24i - 32j - (-24i + 10j)

24i - 32j + 24i - 10j

48i - 42j

So now we have C but we want the direction. First it's important to note which quadrant it will be in. The i direction is positive, so we know it will be right of the y axis, and the j term is negative so we know it will be below the x axis. this is quadrant 4, so we know the answer will be between 270 and 360

It is worth mentioning that arctan only gives aswers from -90 to 90 rather than a full 360,so we should be fine here, the answer will be between -90 and 0 when we take the arctan because that is quadrant 4 as well. if you has something like -5i + 3j you would expect the answer to be in quadrant 2, but you would get an answer in quadrant 4. the trick is to add 180. Similarly if you expect a result in quadrant 3 you will get the answer in quadrant 1. Same deal, just add 180. This is shown in Example 3 in the article

When you want the direction of a vector you take the i term as a and j term as b then take arctan(b/a)

here a = 48 and b = -42 so arctan(-42/48) = -41.19. just to double check this is in quadrant 4 so it is the right answer. If you started with a = -48 and b = 42 you would get the same answer but -48i + 42j is in quadrant 2, so you would take -41.19 and add 180 to get 138.81, which is in quadrant 2.

Let me know if this didn't help(3 votes)

- Do you add 270 degrees to the calculator output in the third quadrant?(3 votes)
- In quadrant 4 why are we adding 360 instead of subtracting 360?(1 vote)
- I'm assuming you are referring to Example 2, where the calculator gives us the value -53 degrees. We must
**add**360 degrees, because that gives us a**positive**value that does not change the angle: -53+360=+307. If we subtracted 360, we would end up with a negative value, and that would not be between 0 and 360, even though we would still have the same angle.(4 votes)

- I wonder how to calcualte the direction of a vector, which is not in the standard position.(1 vote)
- Are there situations in which components are preferable to unit vectors (i-hats and j-hats) and vice versa? And why?(2 votes)