Vectors of magnitude 21 units in the direction of the vector $3 \hat{i} - 6 \hat{j} + 2 \hat{k}$ are _____________.

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Solution

Given: $\vec{a} = 3 \hat{i} - 6 \hat{j} + 2 \hat{k}$

The unit vector in the direction of the given vector $\vec{a}$ is
$\hat{a} = \frac{\vec{a}}{\vec{|a|}} = \frac{3 \hat{i} - 6 \hat{j} + 2 \hat{k}}{\sqrt{{(3)}^{2} + {(- 6)}^{2} + {(2)}^{2}}} = \frac{3 \hat{i} - 6 \hat{j} + 2 \hat{k}}{\sqrt{9 + 36 + 4}} = \frac{3 \hat{i} - 6 \hat{j} + 2 \hat{k}}{\sqrt{49}} = \frac{1}{7} (3 \hat{i} - 6 \hat{j} + 2 \hat{k})$

Therefore, the vector having magnitude 21 units in the direction of $\vec{a}$ is
$\pm 21 \hat{a} = \pm 21 (\frac{1}{7} (3 \hat{i} - 6 \hat{j} + 2 \hat{k})) = \pm 3 (3 \hat{i} - 6 \hat{j} + 2 \hat{k}) = \pm (9 \hat{i} - 18 \hat{j} + 6 \hat{k})$

Hence, vectors of magnitude 21 units in the direction of the vector $3 \hat{i} - 6 \hat{j} + 2 \hat{k}$ are $\pm (9 \hat{i} - 18 \hat{j} + 6 \hat{k}) .$

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3 ˆ i - 6 ˆ j + 2 ˆ k

2 \hat{i} + 3 \hat{j} + 6 \hat{k} and 3 \hat{i} - 6 \hat{j} + 2 \hat{k} .

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