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Vector forms review

Review all the different ways in which we can represent vectors: components, magnitude & direction, and unit vectors.

What are the different vector forms?

Component form(a,b)
Unit vectorsai^+bj^
Magnitude and direction∣∣u∣∣,θ

Component form

In component form, we treat the vector as a point on the coordinate plane, or as a directed line segment on the plane. The components are the vector's x- and y-coordinates.
Want to learn more about vector component form? Check out this video.

Unit vector form

These are the unit vectors in their component form:
i^=(1,0)
j^=(0,1)
Using vector addition and scalar multiplication, we can represent any vector as a combination of the unit vectors. For example, (3,4) can be written as 3i^+4j^.
Want to learn more about unit vectors? Check out this video.

Magnitude and direction form

Considering the vector graphically, we can also describe it by its magnitude (the length of the line segment) and its direction (the angle the line forms with the positive x-axis).
Want to learn more about vector magnitude and direction form? Check out this article.

Want to join the conversation?

  • primosaur ultimate style avatar for user Jordan Cooper
    Are degrees or radians more commonly used for the direction of vectors? And why?
    (23 votes)
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  • aqualine ultimate style avatar for user John He
    I know I can easily transform Component form to Magnitude and direction form.But how can I use Magnitude and direction form to get Component form?
    (10 votes)
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    • duskpin tree style avatar for user budster the dingo
      Let's use vector (3, 4) as an example. That in magnitude direction form is ||5||, 53.130102°.

      Let's think of this vector as a triangle on the unit circle

      First, evaluate what quadrant you're in. This information will judge which sides are negative and which are positive. Now, we need to find the reference angle of our angle. In this case, it's 53.130102°.

      Now, we do some simple trig to find the two sides using our angle and the hypotenuse (the magnitude):

      sin(53.130102) = Opposite / 5
      5 * sin(53.130102) = (Opposite / 5) * 5
      5sin(53.130102) = Opposite
      Opposite = 3.99999998146
      Opposite = 4

      The answers are a bit off because we rounded our angle a couple decimal places

      cos(53.130102) = Adjacent / 5
      5 * cos(53.130102) = (Adjacent / 5) * 5
      5cos(53.130102) = Adjacent
      Adjacent = 3.00000002472
      Adjacent = 3

      Our vector in component form is (3, 4)
      (26 votes)
  • duskpin ultimate style avatar for user blueninja300
    Why are there two different types of measures (degree and radians) rater than 1 standard method of use?
    (2 votes)
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    • aqualine ultimate style avatar for user Rae Riddle
      Because like the Imperial and and Metric systems of measurement, one came first and another, more optimized version was created later. Degrees were invented about 6,000 years ago by the Babylonians, whose love of base 60 and a year lasting approximately 360 days drove them to create an angle measurement system that worked best with both of those. Via cultural diffusion, the degrees system of measurement was passed onto the ancient Egyptians, who used it to create their great pyramids. Radians are much more modern, and relate angle measurements to the length of the arc. Why they generally involve pi, is because pi itself is the ratio between the diameter and circumference of any circle. Because radians are not just arbitrary numbers, and instead fairly concrete mathematical ratios, it is much more useful for higher level mathematics. The first record of a mathematician using this method dates back to 1400 CE, and the term "radian" was first used in 1873 CE. Degrees are still more common in geometry and lower level mathematics because they are simpler and faster to write and understand, and if you live in the USA, you are taught about degrees first. So while radians are mathematically superior, degrees are more practical for everyday use. In theoretical applications of physics, geometry, and calculus, radians are the way to go. If you're designing something to be built or used, degrees are the standard. If you still have questions, I found most of this information online, and you may even find a video here on Khan Academy that helps clarify this for you.
      (19 votes)
  • blobby green style avatar for user Arbaaz Ibrahim
    Can someone please explain magnitude and direction form?
    (4 votes)
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    • aqualine ultimate style avatar for user Rae Riddle
      Magnitude and direction form is seen most often on graphs. You'll see the length of the vector (aka magnitude) written near the graphed vector, and the positive angle it forms with the x-axis as well. It's a fairly clear and visual way to show the magnitude and direction of a vector on a graph.
      (3 votes)