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Q

The value of $\dpi{100} a$, for which the points $\dpi{100} A,B,C$ with position  vectors $\dpi{100} 2\hat{i}-\hat{j}+\hat{k},\; \; \hat{i}-3\hat{j}-5\hat{k}\; \; and\; \; a\hat{i}-3\hat{j}+\hat{k}$  respectively are the vertices of a right angled triangle at $\dpi{100} c$ are

• Option 1)

2 and 1

• Option 2)

–2 and –1

• Option 3)

–2 and 1

• Option 4)

2 and –1

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As we have learned

Scalar Product of two vectors -

$\vec{a}.\vec{b}> 0 \:an\: acute\: angle$

$\vec{a}.\vec{b}< 0 \:an\: obtuse\: angle$

$\vec{a}.\vec{b}= 0 \:a\:right\: angle$

- wherein

$\Theta$  is the angle between the vectors $\vec{a}\:and\:\vec{b}$

Position Vector -

If $\vec{a}$ and $\vec{b}$ are the position of vectors of two points A and B then

$\overrightarrow{AB}= \vec{b}-\vec{a}$

$\overrightarrow{AB}= P \vee of B - P\vee of A$

- wherein

$\vec{AC}\cdot \vec{BC} = 0$

$\Rightarrow ((a\hat{i}-3\hat{j}+\hat{k})-(2\hat{i}-\hat{j}+\hat{k}))\cdot ((a\hat{i}-3\hat{j}+\hat{k})- (\hat{i}-3\hat{j}-5\hat{k}))=0$

( a-2 )( a-1 )= 0

a = 1,2

Option 1)

2 and 1

Option 2)

–2 and –1

Option 3)

–2 and 1

Option 4)

2 and –1

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