The value of ‘a’, for which the points A,B,C with position vector’s 2i¯−j¯+k¯, i¯−3j¯−5k¯, and ai¯−3j¯+k¯ respectively are the vertices of a Right angled triangle with C=π2 are

AC¯=OC¯−OA¯=(a−2)i¯−2j¯BC¯=OC¯−OB¯=(a−1)i¯+6k¯C=π2⇒AC→⋅BC→=0⇒(a−2)(a−1)=0⇒a=2,1

The value of ‘a’, for which the points A,B,C with position vector’s 2i¯−j¯+k¯, i¯−3j¯−5k¯, and ai¯−3j¯+k¯ respectively are the vertices of a Right angled triangle with C=π2 are

AC¯=OC¯−OA¯=(a−2)i¯−2j¯BC¯=OC¯−OB¯=(a−1)i¯+6k¯C=π2⇒AC→⋅BC→=0⇒(a−2)(a−1)=0⇒a=2,1

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The value of ‘a’, for which the points A,B,C with position vector’s $2 \bar{i} - \bar{j} + \bar{k}, \bar{i} - 3 \bar{j} - 5 \bar{k},$ and $a \bar{i} - 3 \bar{j} + \bar{k}$ respectively are the vertices of a Right angled triangle with $C = \frac{π}{2}$ are

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-2 and 1

2 and -1

2 and 1

-2 and -1

answer is C.

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Detailed Solution

$\bar{A C} = \bar{O C} - \bar{O A} = (a - 2) \bar{i} - 2 \bar{j}$
$\begin{array}{l} \bar{B C} = \bar{O C} - \bar{O B} = (a - 1) \bar{i} + 6 \bar{k} \\ C = \frac{π}{2} \Rightarrow \vec{A C} \cdot \vec{B C} = 0 \\ \Rightarrow (a - 2) (a - 1) = 0 \Rightarrow a = 2, 1 \end{array}$