contestada

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.

Respuesta :

LammettHash
ma[tex]k\mathbf u+\mathbf v=k(-1,1,1)+(-7,6,15)=(-k-7,k+6,k+15)[/tex]

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

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

Otras preguntas