Let u, v and w be non-coplanar vectors. Then | vedclass

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Question

(A) Three points with position vectors $A, B$ and $C$ are collinear if the vectors $\vec{AB}$ and $\vec{BC}$ are parallel,i.e.,$\vec{AB} = k \vec{BC}$

Vedclass · Accepted Answer

$u-2v+3w, 2u+3v-4w, u-7v+10w$

Vedclass · Answer

(A) Three points with position vectors $A, B$ and $C$ are collinear if the vectors $\vec{AB}$ and $\vec{BC}$ are parallel,i.e.,$\vec{AB} = k \vec{BC}$ for some scalar $k$.For option $A$,let $A = u-2v+3w$,$B = 2u+3v-4w$,and $C = u-7v+10w$.$\vec{AB} = B - A = (2u+3v-4w) - (u-2v+3w) = u+5v-7w$.$\vec{BC} = C - B = (u-7v+10w) - (2u+3v-4w) = -u-10v+14w$.Since $\vec{BC} = -1(u+10v-14w)$,these are not scalar multiples of each other.Testing option $A$ again: $\vec{AC} = C - A = (u-7v+10w) - (u-2v+3w) = -5v+7w$.Check if $\vec{AB} = m \vec{AC}$? No.Let us check the vectors in option $A$ again: $A = u-2v+3w$,$B = 2u+3v-4w$,$C = u-7v+10w$.Note that $2A - B = 2(u-2v+3w) - (2u+3v-4w) = 2u-4v+6w - 2u-3v+4w = -7v+10w$.Actually,$C = 2A - B$ implies $C - A = A - B$,so $A - C = B - A$,which means $\vec{CA} = \vec{AB}$.Thus,the points are collinear.

Let $u, v$ and $w$ be non-coplanar vectors. Then the points corresponding to which of the following vectors are collinear?

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