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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:
QuadrantHow to adjust
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.
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):
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 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 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 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
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
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
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
space, comma
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
right parenthesis
Round your final answers to the nearest hundredth.

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

Want to join the conversation?

  • leaf green style avatar for user zzzwhking
    problem2.2 has two answers,am i right?
    325.01 & 145.01
    (1 vote)
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    • leaf blue style avatar for user jwinder47
      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)
  • blobby green style avatar for user william_hinkley
    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)
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  • blobby green style avatar for user mesmerjessica
    How do you calculate the component form of a vector in 3D (x,y,z)?
    (3 votes)
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    • female robot grace style avatar for user loumast17
      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)
  • blobby green style avatar for user Aleks Jevtic
    What about 3 Dimensional Vectors? How would I calculate and express the angle it builds with the planes of the origin?
    (3 votes)
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  • blobby green style avatar for user Calvin Haworth
    How do I find which quadrant the vector would be in? I think i missed something along the way.
    (2 votes)
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  • blobby green style avatar for user Yousef Osama
    If vector A = 12i – 16j and vector B = –24i + 10j, what is the direction of the vector C = 2A – B?
    (2 votes)
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    • female robot grace style avatar for user loumast17
      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)
  • mr pants teal style avatar for user jackeames
    Do you add 270 degrees to the calculator output in the third quadrant?
    (3 votes)
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  • female robot grace style avatar for user Wild Chaser
    In quadrant 4 why are we adding 360 instead of subtracting 360?
    (1 vote)
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    • leaf blue style avatar for user jwinder47
      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)
  • blobby green style avatar for user zsoltturi
    I wonder how to calcualte the direction of a vector, which is not in the standard position.
    (1 vote)
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  • primosaur ultimate style avatar for user Jordan Cooper
    Are there situations in which components are preferable to unit vectors (i-hats and j-hats) and vice versa? And why?
    (2 votes)
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