Question #200624

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 >


1
Expert's answer
2021-06-03T16:52:18-0400

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>,kR.<x+6,y+7,z> = k<-1,2,4>, \: k\in \mathbb{R}.

So x=k6,  y=2k7,  z=4k,  kR.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.




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