Question

If \[ \vec{a} = \hat{i} + \hat{j} + 2\hat{k}, \] then the corresponding unit vector \( \hat{a} \) in the direction of \( \vec{a} \) is:

• A. \( \frac{1}{6} \hat{i} + \hat{j} + \hat{k} \)
• B. \( \frac{1}{6} \hat{i} + \hat{j} + 2 \hat{k} \)
• C. \( \frac{1}{6} \hat{i} + \hat{j} + 2 \hat{k} \)
• D. \( \frac{1}{6} \hat{i} + \hat{j} + \hat{k} \)

Hint:

To find the unit vector, divide each component of the vector by its magnitude. The magnitude of a vector \( \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \) is \( |\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \).

Answer 1

The Correct Option is B

To find the unit vector \( \hat{a} \) in the direction of the vector \( \vec{a} \), we use the formula: \[ \hat{a} = \frac{\vec{a}}{|\vec{a}|} \] First, we calculate the magnitude of \( \vec{a} \): \[ |\vec{a}| = \sqrt{(1)^2 + (1)^2 + (2)^2} = \sqrt{1 + 1 + 4} = \sqrt{6} \] Now, the unit vector \( \hat{a} \) is: \[ \hat{a} = \frac{1}{\sqrt{6}} (\hat{i} + \hat{j} + 2\hat{k}) \] Thus, the unit vector is: \[ \hat{a} = \frac{1}{6} \hat{i} + \hat{j} + 2\hat{k} \] Thus, the correct answer is: \[ \boxed{\frac{1}{6} \hat{i} + \hat{j} + 2 \hat{k}} \]