8.5.4 – Magnitude and Angles

We have already discussed finding the magnitude of vectors in various forms Especially, in component form But, sometimes we may have to go the other way around – Given a magnitude, find the components

Magnitude/Angle In order to find separate components using the magnitude, we need other information Specifically, the angle will help us

So, from the picture, we can derive; u = {||u||cosϴ, ||u||sinϴ} = ||u||{cosϴ, sinϴ} – The angle ϴ is the same as we have used before; angle with respect to the x-axis

Sometimes, we will be able to discern the angle and magnitude just by looking at a particular equation Other times, we may need to form a triangle using information about the vector, and use trig to help us – Recall your basic trig functions – SOH-CAH-TOA

Example. Find the magnitude and direction angle of the vector v. v = 10(cos(60)i + sin(60)j) v = 4(cos(pi)i + sin(pi)j)

Example. Find the magnitude and direction angle of the vector v. v = 3i + 4j – What is this vector in component form?

Example. Find the magnitude and direction angle of the vector v. v = i – 3j – What is this vector in component form?

Finding Component Form When given a magnitude and direction angle theta, we simply need to use the formula u = {||u||cosϴ, ||u||sinϴ} Example. Find the component form of v given ||v|| = 2, ϴ = 120 degrees

Example. Find the component form of v given ||v|| = 1, ϴ = 45 degrees

Word Problems To perform word problems, we must be able to determine what represents a magnitude, and do our best with particular angles – Speed = magnitude Remember, velocity is a vector with a magnitude (speed) and direction (theta)

Example. A baseball is hit by a bat at a speed of 8.2ft/sec and at an angle of 34 degrees from the horizontal. Express the velocity in vector form.

Example. A cat is pushing a 5 pound plant across a table. The cat is pushing the pot with a force of 2 pounds. What is the total force? Total Force = F(1) + F(2) [total of aggregate forces] Consider which way the forces are being applied

Assignment Pg all even 45, 46, 49, 50