Find a vector in the direction of the vector 2i^−j^+4k^ which has magnitude 4 units

|A→|=2i^−j^+4k^ |A→|=(2)2+(−1)2+(4)2=21 |A→|=A→|A→|=2i^−j^+4k^21Hence required vector is 4A→=8i^−4j^+16k^21

Find a vector in the direction of the vector 2i^−j^+4k^ which has magnitude 4 units

|A→|=2i^−j^+4k^ |A→|=(2)2+(−1)2+(4)2=21 |A→|=A→|A→|=2i^−j^+4k^21Hence required vector is 4A→=8i^−4j^+16k^21

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Find a vector in the direction of the vector $2 \hat{i} - \hat{j} + 4 \hat{k}$ which has magnitude 4 units

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$\frac{8 \hat{i} - 4 \hat{j} + 16 \hat{k}}{\sqrt{21}}$

$\frac{8 \hat{i} + 4 \hat{j} + 16 \hat{k}}{\sqrt{21}}$

$\frac{8 \hat{i} - 4 \hat{j} + 16 \hat{k}}{\sqrt{5}}$

$\frac{8 \hat{i} - 4 \hat{j} - 16 \hat{k}}{\sqrt{21}}$

answer is C.

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Detailed Solution

$| \vec{A} | = 2 \hat{i} - \hat{j} + 4 \hat{k}$
$| \vec{A} | = \sqrt{{(2)}^{2} + {(- 1)}^{2} + {(4)}^{2}} = \sqrt{21}$
$| \vec{A} | = \frac{\vec{A}}{| \vec{A} |} = \frac{2 \hat{i} - \hat{j} + 4 \hat{k}}{\sqrt{21}}$
Hence required vector is $4 \vec{A} = \frac{8 \hat{i} - 4 \hat{j} + 16 \hat{k}}{\sqrt{21}}$