Using vectors, find the value of such that the points ( , -10, 3)… - Vidyadip

Question

Using vectors, find the value of $\lambda$ such that the points ($\lambda$, -10, 3), (1, -1, 3) and (3, 5, 3) are collinear.

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Answer

Points ($\lambda$, -10, 3), (1, -1, 3) and (3, 5, 3) are collinear.$\therefore$ ($\lambda$, -10, 3) = x(1, -1, 3) + y(3, 5, 3) for some scalars x and y.
$\Rightarrow\lambda$ = x + 3y, -10 = -x + 5y and 3 = 3x +3y
Solving -10 = -x + 5y and 3 = 3x + 3y for x and y we get,
$\text{x}=\frac{5}2$ and $\text{y}=-\frac{3}2$
Now,
$\lambda=\text{x}+3\text{y}$
$\Rightarrow\lambda=\frac{5}2+3\Big(-\frac{3}2\Big)=-2$

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