Decompose vector B into components parallel and perpendicular to C

[hamza324]

Hi , I have a question stated as

Given the vector B=-6x-8y+9z and vector C= 5x-3y+4z .

Find vectors T1 and T2 such that T1 is parallel to vector C and perpendicular to vector T2. where vector B = T1 + T2 .



So far, i was able to find a vector T1 which is parallel to vector C but couldn't figure out how i can make it perpendicular to the vector T2 because when i try to make it perpendicular to the vector T2, it becoms impossible to satisfy the given equation B = T1 + T2 .

 

[Simon Bridge]

Welcome to PF;

Those are equations for lines, not vectors.
Did you mean x,y, and z to be unit vectors?
So $\vec{B}=(-6,-8,9)^t$ and $\vec{C}=(5,-3,4)^t$$\vec{T}_1$ is parallel to $\vec{C}$ and perpendicular to $\vec{T}_2$.
$\vec{B}=\vec{T}_1+\vec{T}_2$

i was able to find a vector $\vec{T}_1$ which is parallel to vector C

There are infinite vectors parallel to $\vec{C}$, but how did you find the particular one you needed?

Per your question:
Hint:

if $\vec{u}\perp\vec{v}$ then $\vec{u}\cdot\vec{v}=?$ and $\vec{u}\times\vec{v}=?$

 

[hamza324]

Thanks a lot...i got it..