Question

The component of \(\hat{i}\) in the direction of the vector \(\hat{i}+\hat{j}+2\hat{k}\) is

• A. \(6\sqrt6\)
• B. \(\sqrt6\)
• C. \(\frac{\sqrt6}{6}\)
• D. 6

Hint:

To find the component of one vector in the direction of another, use the formula \( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \), where \( \vec{a} \) is the vector whose component you want to find, and \( \vec{b} \) is the direction vector.

Answer 1

The Correct Option is C

The correct answer is: (C): \(\frac{\sqrt{6}}{6}\)

We are tasked with finding the component of \( \hat{i} \) in the direction of the vector \( \hat{i} + \hat{j} + 2\hat{k} \).

Step 1: Formula for the component of a vector

The component of a vector \( \vec{a} \) in the direction of a vector \( \vec{b} \) is given by the formula:

\( \text{Component of } \vec{a} \text{ in the direction of } \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \)

Step 2: Compute the dot product

Let \( \vec{a} = \hat{i} \) and \( \vec{b} = \hat{i} + \hat{j} + 2\hat{k} \). The dot product \( \vec{a} \cdot \vec{b} \) is:

\( \vec{a} \cdot \vec{b} = (1)(1) + (0)(1) + (0)(2) = 1 \)

Step 3: Find the magnitude of \( \vec{b} \)

The magnitude of vector \( \vec{b} \) is:

\( |\vec{b}| = \sqrt{1^2 + 1^2 + 2^2} = \sqrt{6} \)

Step 4: Compute the component

Now we can compute the component of \( \hat{i} \) in the direction of \( \vec{b} \) using the formula:

\( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} = \frac{1}{\sqrt{6}} = \frac{\sqrt{6}}{6} \)

Conclusion:
The component of \( \hat{i} \) in the direction of \( \hat{i} + \hat{j} + 2\hat{k} \) is \( \frac{\sqrt{6}}{6} \), so the correct answer is (C): \(\frac{\sqrt{6}}{6}\).