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Algebra of Vectors — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors3 Marks
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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