Question

Let A = (1, 0, 3) and B = (−3, 2, 1).

Answer 1

Answer:

a) $(-\frac{\sqrt{24}}{6} ,\frac{\sqrt{24}}{12} ,-\frac{\sqrt{24}}{12})$

b) $(-1,1,2)$

c) $-2x+y-z=1$

Step-by-step explanation:

a)

AB, OB, OA are vectors.

$AB = OB - OA = (-3, 2, 1) - (1, 0, 3) = (-4, 2, -2)\\||AB|| = \sqrt{(-4)^2+2^2+(-2)^2}=\sqrt{24}$

Normalizing AB: $\frac{AB}{||AB||} = (\frac{-4}{\sqrt{24}},\frac{2}{\sqrt{24}}, \frac{-2}{\sqrt{24}})=(-\frac{\sqrt{24}}{6} ,\frac{\sqrt{24}}{12} ,-\frac{\sqrt{24}}{12})$

b)

OM, OA, OB are vectors.

$OM = \frac{1}{2}OA+\frac{1}{2}OB=(\frac{-3+1}{2} ,\frac{2+0}{2}, \frac{1+3}{2} )=(-1,1,2)$

So, $M = (-1,1,2)$

c)

$-4x+2y-2z=(-4)(-1)+(2)(1)+(-2)(2)=4+2-4=2$

Hence,

$-2x+y-z=1$