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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

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?

Similar Questions

Let $a = i$ be a vector which makes an angle of $120^\circ$ with a unit vector $b$. Then the unit vector $(a + b)$ is

The position vectors of four points $P, Q, R, S$ are $2a + 4c$,$5a + 3\sqrt{3}b + 4c$,$-2\sqrt{3}b + c$,and $2a + c$ respectively. Then:

If $\hat{x}, \hat{y},$ and $\hat{z}$ are three unit vectors in three-dimensional space,then find the minimum value of $|\hat{x} + \hat{y}|^2 + |\hat{y} + \hat{z}|^2 + |\hat{z} + \hat{x}|^2$.

If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors,then $|\vec{b}| \vec{a} + |\vec{a}| \vec{b}$ represents:

Show that the vector $\hat{i}+\hat{j}+\hat{k}$ is equally inclined to the axes $OX, OY,$ and $OZ$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module