F = 3x-y+2z in newtons
the line that it follows is -x+y+2z
What is the work done
The Attempt at a Solution
I am guessing
One way to evaluate the dot product of two vectors A and B (boldface denotes vectors) is to evaluate it component-wise. In other words, if:I would assume that the x, y, and z are vectors.
As to the dot product, I have no clue. It would have been nice if it was ever covered....
Well, unless if x, y and z are unit vectors (i.e. they correspond to what I called i, j, and k), your vectors don't really make sense.So Work = -3x-y+4z
I had part of a class way back in Trig that vaguely talked about vector multiplication but since it was the day before the final it was considered a "bonus" lesson. Yeah, my physics professor, and the word teach don't exactly mix.
Or do the alpha characters just go away, and thus I would have zero as the answer
That's exactly right. Take another look at the very last equation in my previous post, and you'll see that it says the following. To compute the dot product of the two vectors:If, and I do stress the IF I am understanding your explanation of the dot product then it would follow like this:
Force = 3x-y+2z (technically the way it was stated is "component 3 is in newtons")
Line = -x +y+2z (also stated as "- is in meters")
Find the work, ergo W=F*d
I am assuming that when my professor put line, he meant straight, and that was the distance.
Therefore it should look something like
Force * line = work
which equals zero
Okay, then this all makes perfect sense. Those are UNIT VECTORS. In a Cartesian coordinate system, any vector can be resolved into three components. If you look carefully at the expression, you'll see that what it is saying is that the vector is the sum of three individual vectors, one of which points entirely in the x-direction, one of which points entirely in the y-direction, and one of which points entirely in the z-direction. Each "unit vector" has the corresponding component as its coefficient. In other words, the unit vectors (which have length 1) have each been "scaled" to the right length for that component of the vector. That's what the notation means.P.S. all of the x, y, and z had the "carat" on top of them.