Question

Let r3 have the euclidean inner product. let u = (-1, 1, 1) and v = (-7, 6, 15). if ||ku + v||= 7, what is k? give the exact answer using fractions if necessary. give the answers in descending order.

Answer 1

ma$k\mathbf u+\mathbf v=k(-1,1,1)+(-7,6,15)=(-k-7,k+6,k+15)$

Recall that for a vector $\mathbf x\in\mathbb R^n$, we have $\|\mathbf x\|=\sqrt{\mathbf x\cdot\mathbf x}$. So we have

$\|k\mathbf u+\mathbf v\|=\sqrt{(k\mathbf u+\mathbf v)(k\mathbf u+\mathbf v)}=7$
$\implies k^2\mathbf u\cdot\mathbf u+2k\mathbf u\cdot\mathbf v+\mathbf v\cdot\mathbf v=49$
$\implies3k^2+56k+261=0$
$\implies k=-9,-\dfrac{29}3$