Let a= i+ j+ k, b=3 i-5 j+4 k, with a b=- i+9 j+12 k.

Question

Let $\vec a=\alpha\hat i+\hat j+\beta\hat k$, $\vec b=3\hat i-5\hat j+4\hat k$, with $\vec a	imes\vec b=-\hat i+9\hat j+12\hat k$. The projection of $\vec b-2\vec a$ on $\vec b+\vec a$ is:

Accepted Answer

$$\vec a\times\vec b = \begin{vmatrix}\hat i&\hat j&\hat k\ \alpha&1&\beta\3&-5&4\end{vmatrix} = (4+5\beta)\hat i+(3\beta-4\alpha)\hat j+(-5\alpha-3)\hat k = (-1,9,12)$$

$4+5\beta=-1\Rightarrow\beta=-1$; $-5\alpha-3=12\Rightarrow\alpha=-3$.

$\vec b-2\vec a = (3,-5,4)-2(-3,1,-1) = (9,-7,6)$; $\vec b+\vec a=(0,-4,3)$.

$$\text{Projection} = \frac{(9)(0)+(-7)(-4)+(6)(3)}{|\vec b+\vec a|} = \frac{28+18}5 = \frac{46}5$$

Solutions · 1