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\|=√(\mathbf x\cdot\mathbf x)`. So we have


`\|k\mathbf u+\mathbf v\|=√((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,-\frac{29}3`