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  4. If the points with position vectors 10 hati + 3 hatj , 12 hati - 5 hatj and x hati + 11 hatj are collinear, then x =

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Q. If the points with position vectors $10\hat{i} + 3\hat{j} , 12 \hat{i} - 5\hat{j}$ and $x\hat{i} + 11 \hat{j}$ are collinear, then $x$ =

COMEDKCOMEDK 2010Vector Algebra

Solution:

The vectors $10\hat{i} + 3\hat{j} , 12 \hat{i} - 5\hat{j}$ and $x\hat{i} + 11 \hat{j}$ are collinear.
$ \therefore \:\:\:\:\Delta =\begin{vmatrix}10&3&1\\ 12&-5&1\\ x&11&1\end{vmatrix} = 0 $
$\Rightarrow 10\left(-5-11\right)-3\left(12-x\right)+1\left(132+5x\right)=0 $
$\Rightarrow -160-36 +3x+132+5x=0$
$ \Rightarrow 8x -+ 4 =0 \Rightarrow x = 8$