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Q.
If three points A, B and $C$ have position vectors $(1, x, 3)$, $(3,4,7)$ and $(y,-2,-5)$ respectively and, if they are collinear, then $( x ,- y )$ is equal to
Vector Algebra
Solution:
Given that $\overrightarrow{ OA }=\hat{ i }+\hat{ x }+3 \hat{ k }$
$\overrightarrow{ OB }=3 \hat{ i }+4 \hat{ j }+7 \hat{ k }$ and $\overrightarrow{ OC }= y \hat{ i }-2 \hat{ j }-5 \hat{ k }$
Since, $A, B, C$ are collinear, Then, $\overrightarrow{ AB }=\lambda \overrightarrow{B C}$
$\Rightarrow 2 \hat{i}+(4-x) \hat{j}+4 \hat{k}=\lambda[(y-3) \hat{i}-6 \hat{j}-12 \hat{k}]$
On comparing the coefficients of $\hat{ i }, \hat{ j }$ and $\hat{ k }$,we get
$2=( y -3) \lambda\,\,\,\,\, \dots(i)$
$4-x=-6 \lambda\,\,\,\,\,\dots(ii)$
and $4=-12 \lambda$
$ \Rightarrow \lambda=-\frac{1}{3}\,\,\,\,\,\,\dots(iii)$
On putting the value of $\lambda$ in eqs. (i) and (ii),we get
$y=-3$ and $x=2$