A vector in two dimensions has components in both directions. Here’s how to add vectors together to get a single resultant vector using component addition as well as trigonometry (the law of cosines and the law of sines):
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It’s part of the original problem. This would be like a force pushing on a box. I am showing you how to handle forces when they are along the negative axes.
The MAGNITUDE (5N) is the same for both because it’s the same force. We’re breaking that force at an angle into it’s x-dir and y-dir components.
You can write the force vector as a magnitude plus direction (that’s what a vector is), like 10N at 225 degrees. Or you can write it as x and y components as shown in the video.
If you just write “10N” without any direction, that’s called a ‘scalar’ quantity.
55 mph is a scalar. 55 mph NW is a vector. Do you see the difference?
Starting from 4:47, you list the force in quadrant II as 5N, but I’m not sure about how you determined both -5 in the x direction and +5 in the y direction. Is there a special relationship I can refer to?