Vector Components, Magnitude, and Direction
Date: 07/23/98 at 06:44:19 From: Kristine Tan Subject: Physics (a vector problem) Vector M of magnitude 4.75m is at 58.0 degrees counter-clockwise from the positive x-axis. It is added to vector N, and the resultant is a vector of magnitude 4.75m, at 39 degrees counterclockwise from the positive x-axis. Find: (a) the components of N, and (b) the magnitude and direction of N. I drew a graphical illustration of the problem. But I really can't solve it because I don't know how. Please help. Thank you.
Date: 07/23/98 at 11:57:54 From: Doctor Rick Subject: Re: Physics (a vector problem) Hi, Kristine, I will get you started on solving this kind of problem. There are two tools you need to do this: (1) converting between magnitude/direction and components of a vector and (2) adding vectors. The first requires some trigonometry, so I hope you've had some. (1) You are given the magnitude and direction of vectors M and P (the sum of M and N). Before you can add them, you must find their components. Remember this diagram: My+-------------* M | /| | / | | / | | / | | / | | L/ | | / | sin(a)|-----+ | | /| | | 1/ | | | / | | | /)a | | |/____|_______|__________ O cos(a) Mx A vector of length 1 has components (cos(a), sin(a)). By similar triangles, a vector M of length L has components Mx = L*cos(a), My = L*sin(a). Do this with both vectors M and P to get their components (Mx, My) and (Px, Py). (2) You know that M + N = P. To add vectors, add their components: Mx + Nx = Px My + Ny = Py You know Mx, My, Px, and Py, so you should be able to figure out Nx and Ny. These are the components of vector N. (3) You were also asked for the magnitude and direction of vector N. To do this, you have to reverse step 1. Here's how, using the figure (remember, you'll be doing this for vector N, not vector M). Magnitude(M) = Mx^2 + My^2 (the Pythagorean Theorem where ^2 means square) tangent(a) = sin(a)/cos(a) = My/Mx (by similar triangles again) So Direction(M) = a = inverse tangent of a = arctan(a) Those are the tools you'll need. See if you can do the job now. Write back if you're still confused after you've tried it. - Doctor Rick, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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