Answer to Question #200624 in Linear Algebra for Vina

Question

Find a nonzero vector ~u with initial point P(−6, −7, 0) such that

(1.1) ~u has the same direction as ~v =< −1, 2, 4 >.

(1.2) ~u is oppositely directed to ~v =< −1, 2, 4 >

Answer 1

Let the coordinates of the end of the vector be (x, y, z). Therefore, if we move the initial point to the origin, the coordinates of vector will be <x+6, y+7, z>

A) if this vector is parallel to <-1,2,4>, then $<x+6,y+7,z> = k<-1,2,4>, \: k\in \mathbb{R}.$

So $x= - k-6, \; y =2k-7, \; z=4k, \; k\in \mathbb{R}.$

We may choose k=1 for vector to be parallel and to have the same direction and obtain x=-7, y=-5, z=4.

B) for vector to have the opposite direction we may choose k=-1 and obtain x=-5, y=-9, z=-4.