Answer to Question #104502 in Linear Algebra for havefun7741

Question

Find a non-zero unit vector u with terminal point Q(-7,4,4) such that u has the opposite direction as v=(3,2,1)

Answer 1

Answer: PQ(3t-7,2t+4,t+4)

Let the initial point $P$ of vector $u~$be $(x, y, z)$, and $Q(-7,4,4)$, thus for

$u=PQ,$ where coordinates of $PQ$ are computed as the difference of corresponding coordinates of $Q$ and $P$ , we get

$u=(-7-x,4-y,4-z).$ Vectors in the opposite direction must be (negative) scalar multiples of each other. So we need $PQ=-t\cdot\upsilon$, where $t$ is a scalar $(>0)$.

There is an infinite number of choices; let's pick $t$ , then $PQ=-t\cdot\upsilon$, where $\upsilon=(3,2,1)$,

so $-7-x=-3t, 4-y=-2t, 4-z=-t$

OR: $x = 3t-7, y = 2t+4, z =t+4.$ So vector $PQ$ with initial points $P(3t-7, 2t+4, t+4)$ will be in the opposite direction to $\upsilon$ .