Express the vector →u below as a sum of two vectors →u1 and →u2, where →u1 is parallel to the vector →v given below, and →u2 is perpendicular to →v. Make sure that the first vector in your sum is →u1 and the second is →u2. Use the square root symbol '√' where needed to give an exact value for your answer.

Question

→v = (3,−1,1)
→u = (−1,−6,−3) = (0,0,0) + (0,0,0) = →u1+→u2
Can someone please help me, I was at this for about an hour..

Answer 1

can someone please help..

Answer 2

Dang 4 years later I'm stuck on the same question and still no response :(

Answer 3

Ok so I finally got a way to do it!
So something that is parallel to v will be :
k[x1, y1, z1] --> k being a constant multiplied by vector v, x1 y1 z1 are coordinates of v

So how do we find k?
Since the sums have to equal to u, we know that u - u1 = u2 (our first equation)
Therefore set up the equation so that we have :
[u - kx, u - ky, u - kz] = u2 --> x y z being the coordinates of u1

Ok now moving on..
u2 is perpendicular to v meaning the dot product is zero
So: u2 (dot) v = 0 (our second equation)
We already know what u2 is from the first equation, plug that in and we get..
x1(u - kx) + y1(u - ky) + z1(u-kz) = 0 --> x1 y1 z1 being the coordinates of v and x y z being the coordinates of u1

After solving for k we substitute that into k[x1, y1, z1] = u1
After solving for u1 we plug that into our first equation and solve for u2

ps. there are some helpful videos on youtube as well!

Answer 4

First you have find the projection of u onto v to find u1. then you can simple subtract u from projection of u onto v to get your u2.