Hey professor. I solved this problem, but I wanted to understand it a little bit more. Because I tried earlier to just multiply the vector of <4,-4,1> by 2, giving the vector <8,-8,2>. Since it's just a scaled vector, wouldn't it still be pointing in the same direction? Why would 4/(\sqrt(33)) be the answer instead of 8? Does scaling it by 2 make the vector point in a different direction?
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Ok, as discussed in the lecture last Thursday and in section 10.2 of the textbook (the top of p. 548 in my copy), a unit vector is a vector of length equal to 1. The way you get a Unit vector u^ pointing in the same direction as a vector u is by multiply u by one over it's magnitude. Since ||<4,-4,1>||=√[4^2+(-4)^2+1^2]=√(16+16+1)=√33, the unit vector is
u^=(1/√33)u=(1/√33)<4,-4,1>=<4/√33,-4/√33,1/√33>
To a couple other points of confusion: yes, multiplying a vector by *ANY* positive number gives you another vector pointing in the same direction. Since the magnitude of u is √33, though, 2u has a magnitude of 2√33, which is not 1, so 2u is not a unit vector.
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