Which of the following unit vector is coplanar with vectorsA-2i-3j-k and B-3i-j-3k and orthogonal to the vector C-8i-5j-2k

The unit vector which is coplanar with the vectors nbsp;A→ nbsp;and nbsp;B→ nbsp;and perpendicular to the vector nbsp;C→ nbsp;will be along the dire ...

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Standard XI

Mathematics

Vector Triple Product

Which of the ...

Question

Which of the following unit vector is coplanar with vectors
$\to A = 2 ˆ i - 3 ˆ j + ˆ k and \to B = 3 ˆ i - ˆ j - 3 ˆ k$
and orthogonal to the vector
$\to C = 8 ˆ i - 5 ˆ j + 2 ˆ k$

- \frac{145 ˆ i - 102 ˆ j + 122 ˆ k}{\sqrt{{(145)}^{2} - {(102)}^{2} + {(122)}^{2}}}

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\frac{53 ˆ i + 36 ˆ j - 122 ˆ k}{\sqrt{18989}}

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\frac{145 ˆ i - 102 ˆ j + 122 ˆ k}{\sqrt{{(145)}^{2} - {(102)}^{2} + {(122)}^{2}}}

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- \frac{53 ˆ i + 36 ˆ j - 122 ˆ k}{\sqrt{18989}}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

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Solution

The correct option is D $- \frac{53 ˆ i + 36 ˆ j - 122 ˆ k}{\sqrt{18989}}$
The unit vector which is coplanar with the vectors $\to A and \to B$ and perpendicular to the vector $\to C$ will be along the direction of vector triple product of $\to A, \to B and \to C$ .
$λ (\to A \times \to B) \times \to C = λ [(\to A . \to C) \to B - (\to B . \to C) \to A]$
$\to A . \to C = (2 ˆ i - 3 ˆ j + ˆ k) . (8 ˆ i - 5 ˆ j + 2 ˆ k) = 33 \to B . \to C = (3 ˆ i - ˆ j - 3 ˆ k) . (8 ˆ i - 5 ˆ j + 2 ˆ k) = 23$
$∴ λ (\to A \times \to B) \times \to C = λ (33 (3 ˆ i - ˆ j - 3 ˆ k) - 23 (2 ˆ i - 3 ˆ j + ˆ k)]$

$= λ (53 ˆ i + 36 ˆ j - 122 ˆ k]$

Unit vector will be

$= \pm \frac{53 ˆ i + 36 ˆ j - 122 ˆ k}{\sqrt{{(53)}^{2} + {(36)}^{2} + {(- 122)}^{2}}}$

$= \pm \frac{53 ˆ i + 36 ˆ j - 122 ˆ k}{\sqrt{18989}}$

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Similar questions

Q. Which of the following unit vector is coplanar with vectors

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and orthogonal to the vector

\to C = 8 ˆ i - 5 ˆ j + 2 ˆ k

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Which of the following unit vector is coplanar with vectors →A=2ˆi−3ˆj+ˆk and →B=3ˆi−ˆj−3ˆk and orthogonal to the vector →C=8ˆi−5ˆj+2ˆk

Which of the following unit vector is coplanar with vectors
$\to A = 2 ˆ i - 3 ˆ j + ˆ k and \to B = 3 ˆ i - ˆ j - 3 ˆ k$
and orthogonal to the vector
$\to C = 8 ˆ i - 5 ˆ j + 2 ˆ k$