Question 1

Are degrees or radians more commonly used for the direction of vectors? And why?

Accepted Answer

i think, radians are more commonly used, because the radians has the relationship between arc length and radius.

Question 2

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?

Accepted Answer

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)
```

Question 3

Can someone please explain magnitude and direction form?

Accepted Answer

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.

Question 4

Why are there two different types of measures (degree and radians) rater than 1 standard method of use?

Accepted Answer

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.


Component form	$(a, b)$ ‍
Unit vectors	$a \hat{i} + b \hat{j}$ ‍
Magnitude and direction	$∣∣ \vec{u} ∣∣, θ$ ‍

Course: Algebra (all content) > Unit 19

Vector forms review

What are the different vector forms?

Component form

Unit vector form

Magnitude and direction form

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