6.

# Component Form

If we are given a vector's magnitude and its angle, then we can use Trigonometric Functions to write its components. If the magnitude is \(v\) and the angle is \(\theta\), then the components will always have one of two forms: $$(\pm v\cos\theta, \pm v\sin\theta) \qquad \hbox{or} \qquad (\pm v\sin\theta, \pm v \cos \theta)$$Here's how you do it, in the conventional basis:

- Find the magnitude of the vector (it might be a number or a variable), and write it in both places. Write the angle in both places as well, leaving room for "sin" or "cos".
- Figure out which direction the angle is measured from. If it is measured from the horizontal axis, then the first term gets the cosine, and the second gets the sine. If it is measured from the vertical axis, then the second term gets the cosine, and the first gets the sine.
- If the vector generally points to the right, then the first term gets a +; to the left, it gets a \(-\).
- Similarly, if the vector generally points upward, the second term gets a +; downward, \(-\).

Get used to saying

"The angle is measured from the \(x\), so the \(x\) gets the cosine."

Here are a couple examples.

### Example

- The magnitude is 4 and the angle is 30°, so we fill them both in.
- The angle is measured from the \(x\) axis, so the \(x\) term gets the cosine.
- The vector is pointing to the right, so the first term is positive.
- The vector is pointing upward, so the second term is positive as well.

Thus, \((+4\cos 30\deg,+4\sin 30\deg)\). (I recommend *always* including the sign in vector tuples, even when they are positive.)

See if you can follow the same reasoning for this vector: