Question:
The dot product of \((\mathbf{i}+\mathbf{j}+\mathbf{k})\) and the unit vector parallel to \((i-j+k) is:
The dot product of \((\mathbf{i}+\mathbf{j}+\mathbf{k})\) and the unit vector parallel to \((i-j+k) is:
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Unit vector is always obtained by dividing a vector by its magnitude before performing dot product calculations.
Updated On: Jun 19, 2026
- \(\frac{1}{\sqrt{3}}\)
- \(\frac{2}{\sqrt{3}}\)
- \(\sqrt{3}\)
- \(3\)
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The Correct Option is A
Solution and Explanation
Step 1: Understand the given vectors.
We are given: \[ \vec{A} = \mathbf{i} + \mathbf{j} + \mathbf{k} \] and another vector: \[ \vec{B} = \mathbf{i} - \mathbf{j} + \mathbf{k} \] We need the dot product of \(\vec{A}\) with the unit vector along \(\vec{B}\).
Step 2: Find magnitude of vector \(\vec{B}\).
\[ |\vec{B}| = \sqrt{1^2 + (-1)^2 + 1^2} = \sqrt{3} \]
Step 3: Find unit vector along \(\vec{B}\).
\[ \hat{B} = \frac{\vec{B}}{|\vec{B}|} = \frac{\mathbf{i} - \mathbf{j} + \mathbf{k}}{\sqrt{3}} \]
Step 4: Compute dot product.
\[ \vec{A} \cdot \hat{B} = (\mathbf{i}+\mathbf{j}+\mathbf{k}) \cdot \frac{(\mathbf{i}-\mathbf{j}+\mathbf{k})}{\sqrt{3}} \] \[ = \frac{1 - 1 + 1}{\sqrt{3}} \]
Step 5: Simplify expression.
\[ = \frac{1}{\sqrt{3}} \]
Step 6: Final conclusion.
Thus, the required dot product is: \[ \frac{1}{\sqrt{3}} \]
Final Answer: \[ \boxed{\frac{1}{\sqrt{3}}} \]
We are given: \[ \vec{A} = \mathbf{i} + \mathbf{j} + \mathbf{k} \] and another vector: \[ \vec{B} = \mathbf{i} - \mathbf{j} + \mathbf{k} \] We need the dot product of \(\vec{A}\) with the unit vector along \(\vec{B}\).
Step 2: Find magnitude of vector \(\vec{B}\).
\[ |\vec{B}| = \sqrt{1^2 + (-1)^2 + 1^2} = \sqrt{3} \]
Step 3: Find unit vector along \(\vec{B}\).
\[ \hat{B} = \frac{\vec{B}}{|\vec{B}|} = \frac{\mathbf{i} - \mathbf{j} + \mathbf{k}}{\sqrt{3}} \]
Step 4: Compute dot product.
\[ \vec{A} \cdot \hat{B} = (\mathbf{i}+\mathbf{j}+\mathbf{k}) \cdot \frac{(\mathbf{i}-\mathbf{j}+\mathbf{k})}{\sqrt{3}} \] \[ = \frac{1 - 1 + 1}{\sqrt{3}} \]
Step 5: Simplify expression.
\[ = \frac{1}{\sqrt{3}} \]
Step 6: Final conclusion.
Thus, the required dot product is: \[ \frac{1}{\sqrt{3}} \]
Final Answer: \[ \boxed{\frac{1}{\sqrt{3}}} \]
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