Find Vector C when given Vector A & B.

[ichivictus]

This is from Schaum's 3000 solved Physics problems (1.75). The teacher attempted to guide us to solve it but me and a few classmates are still struggling to figure this out.

Homework Statement

Vector A = 3i + 5j - 2k
Vector B = -3j + 6k
Find a Vector C such that 2A + 7B + 4C = 0

The Attempt at a Solution

This looks a lot like Linear Algebra, something I am not particularly skilled in, however I think I gave it a decent shot.

Ax = 3
Ay = 5
Az = -2

Bx = -3
Bz = 6

2(3) + 7(-3) + 4(Cx) = 0
Cx = 15/4 = 3.75

2(5) + 0 + 4(Cy) = 0
Cy = -10/4 = -5/2 = -2.5

2(-2) + 7(6) + 4(Cz) = 0
Cz = -38/4 = -19/2 = -9.5

So therefor Vector C = (15/4)i - (5/2)j - (19/2)k

The real solution is C = -1.5i + 2.75j - 9.5k

Looks like I got Cz correct, but I can't figure out how to get Cx and Cy.

 

[jedishrfu]

First off your B components are wrong Bx=0 By=-3 and Bz=6

and you have three equations to get the three C components:

2Ax + 7Bx + 4Cx = 0

2Ay + 7By + 4Cy = 0

2Az + 7Bz + 4Cz = 0

so its simple algebra from here

 

[CWatters]

Ax = 3
Ay = 5
Az = -2

Bx = -3
Bz = 6

Should be By=-3 not Bx ?

Edit: jedishrfu beat me to it.

 

[ichivictus]

Oh simple mistake ha. Thanks!

 

[HalfLostAndSearching]

Can anyone please provide some guidance or a solution to this problem?

I can understand your struggle with this problem. It is important to remember that in vector addition, the components of each vector must be added separately. In this case, we have 2A + 7B + 4C = 0, so we can rewrite this as 2(3i + 5j - 2k) + 7(-3j + 6k) + 4Cx + 4Cy + 4Cz = 0. This gives us:

6i + 10j - 4k - 21j + 42k + 4Cx + 4Cy + 4Cz = 0

Combining like terms, we get:

(6 + 4Cx)i + (-11 + 4Cy)j + (38 + 4Cz)k = 0

This means that:

6 + 4Cx = 0
-11 + 4Cy = 0
38 + 4Cz = 0

Solving these equations, we get:

Cx = -1.5
Cy = 2.75
Cz = -9.5

Therefore, Vector C = -1.5i + 2.75j - 9.5k. I hope this helps you understand the solution to this problem. Remember to always pay attention to the components of each vector and add them separately. Good luck with your studies!