How to find two unknown components of a vector

Question

I have found the length of AB to be 60m and the angle between AB - BC is 64 degrees but I am unable to find the length of the vector BC with two unknown vector components. I am unable to find much help on the internet.

Thanks

Answer 1

Hint:

You have for the equation of the line through $B$ and $C$: $$(40,0,-20) + t(3,4,1) = (a,b,0)$$

Now, solve the equation for $t$ using the $z$-coordinate.

Answer 2

(c) the vector equation of BC
$r=\begin{pmatrix} 40\\0\\-20 \end{pmatrix} + \lambda\begin{pmatrix} 3 \\4\\1\end{pmatrix}$ passes through (a,b,$0$)
$\because z=0 \space then \space 0=-20+\lambda\\ \therefore\lambda=20$
$ a=40+20 \times 3\\ \space \space=100\\ b=0+20 \times 4\\ \space \space=80$