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Equation of Perpendicular from a Point on a Line

The point whe...

Question

The point where $\to x$ which is perpendicular to $(2, - 3, 1)$ and $(1, - 2, 3)$ and which satisfies the condition $\to x \cdot (^i + 2^j - 7^k) = 10$

(3, 5, 1)

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(7, - 5, 1)

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(3, - 5, 1)

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(7, 5, 1)

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Solution

The correct option is A $(7, 5, 1)$
$¯ x = (a, b, c)$
$¯ x ⊥ (2, - 3, 1) & x ⊥ (1, - 2, 3)$
$\Rightarrow x . (2, - 3, 1) = 0$ & $x . (1, - 2, 3) = 0$
$\Rightarrow 2 a - 3 b + c = 0$ (1) & $a - 2 b + 3 c = 0$ (2)
$¯ x . (^i + 2^j - 7^k) = 10$
$\Rightarrow a + 2 b - 7 c = 10$ (3)
$\begin{matrix} (1) \Rightarrow c = 3 b - 2 a (2) \Rightarrow a = 2 b - 3 c \end{matrix}}$ Put in (3)
$a + 2 b - 7 c = 10$
$2 b - 3 c + 2 b - 7 c = 10$
$\Rightarrow 4 b - 10 c = 10$ (4)
(1) $\Rightarrow a = \frac{- c + 3 b}{2}$
$a = \frac{3 b - c}{2} = 2 b - 3 c$
$\Rightarrow 3 b - c = 4 b - 6 c$
$\Rightarrow b = 5 c$ __(5)
from (4) & (5)
$c = 1, b = 5, a = 7$
$\Rightarrow ¯ x = (7, 5, 1)$

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The point where →x which is perpendicular to (2,−3,1) and (1,−2,3) and which satisfies the condition →x⋅(^i+2^j−7^k)=10

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