Question

Suppose a car is traveling at 55 km/hr, and that the positive y-axis is north and the positive x-axis is east. Resolve the car's velocity vector (in 2-space) into components if the car is traveling in each of the following directions
east:
south:
southeast:
northwest:

Answer 1

\[\mathbf v = \langle v_x,v_y\rangle
\]And \[
\|\mathbf v \| = 55
\]

Answer 2

when it is going east, all of the magnitude is in the x direction, so: \[
\mathbf v = \langle 55,0 \rangle
\]

Answer 3

When going south, the magnitude is opposite of the y direction so:\[
\mathbf v = \langle 0, -55 \rangle
\]

Answer 4

This is an example of northeast. Basically you project the vector onto it's component. You can use trig functions to do this for any angle, but in this special case where the angle is 45 degrees, you can just use basic algebra to get the \(r/\sqrt2\)

Answer 5

So for south easy, the x is positive and the y is negative: \[
\mathbf v = \left\langle \frac{55}{\sqrt 2}, -\frac{55}{\sqrt 2} \right\rangle
\]

Answer 6

for north west, the x is negative and the y is positive:\[

\mathbf v = \left\langle -\frac{55}{\sqrt 2}, \frac{55}{\sqrt 2} \right\rangle
\]