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

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.

→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

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 2

can someone please help..

Answer 3

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

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.

Answer 5

To express vector →u as a sum of two vectors →u1 and →u2, where →u1 is parallel to →v and →u2 is perpendicular to →v, we need to find scalar multiples of →v that add up to →u.

First, let's find →u1, the vector parallel to →v. To find this, we use the dot product between →u and →v divided by the magnitude of →v squared. The formula for →u1 is:

→u1 = ((→u · →v) / (|→v|^2)) * →v

Let's calculate →u1:

→u · →v = (-1 * 3) + (-6 * -1) + (-3 * 1) = -3 + 6 - 3 = 0
|→v|^2 = 3^2 + (-1)^2 + 1^2 = 9 + 1 + 1 = 11

→u1 = ((→u · →v) / (|→v|^2)) * →v = (0 / 11) * (3, -1, 1) = (0, 0, 0)

Next, let's find →u2, the vector perpendicular to →v. To find this, we subtract →u1 from →u:

→u2 = →u - →u1

→u2 = (−1,−6,−3) - (0, 0, 0) = (-1, -6, -3)

So, we have expressed →u as a sum of →u1 and →u2:

→u = →u1 + →u2 = (0, 0, 0) + (-1, -6, -3) = (-1, -6, -3)

Therefore, →u can be expressed as the sum of →u1 = (0, 0, 0) and →u2 = (-1, -6, -3).