Question:
If the scalar projection of the vector \(x\mathbf{i} - \mathbf{j} + \mathbf{k}\) on the vector \(2\mathbf{i} - \mathbf{j} + 5\mathbf{k}\) is \(\frac{1}{\sqrt{30}}\) then value of \(x\) is equal to
If the scalar projection of the vector \(x\mathbf{i} - \mathbf{j} + \mathbf{k}\) on the vector \(2\mathbf{i} - \mathbf{j} + 5\mathbf{k}\) is \(\frac{1}{\sqrt{30}}\) then value of \(x\) is equal to
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Scalar projection = \(\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}\).
Updated On: Apr 23, 2026
- \(\frac{-5}{2}\)
- 6
- -6
- 3
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The Correct Option is A
Solution and Explanation
Step 1: Formula / Definition}
\[ \text{Projection} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} \]
Step 2: Calculation / Simplification}
\(\mathbf{a} = (x, -1, 1)\), \(\mathbf{b} = (2, -1, 5)\)
\(\mathbf{a} \cdot \mathbf{b} = 2x + 1 + 5 = 2x + 6\)
\(|\mathbf{b}| = \sqrt{4 + 1 + 25} = \sqrt{30}\)
\(\frac{2x+6}{\sqrt{30}} = \frac{1}{\sqrt{30}} \Rightarrow 2x + 6 = 1\)
\(2x = -5 \Rightarrow x = -\frac{5}{2}\)
Step 3: Final Answer
\[ -\frac{5}{2} \]
\[ \text{Projection} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} \]
Step 2: Calculation / Simplification}
\(\mathbf{a} = (x, -1, 1)\), \(\mathbf{b} = (2, -1, 5)\)
\(\mathbf{a} \cdot \mathbf{b} = 2x + 1 + 5 = 2x + 6\)
\(|\mathbf{b}| = \sqrt{4 + 1 + 25} = \sqrt{30}\)
\(\frac{2x+6}{\sqrt{30}} = \frac{1}{\sqrt{30}} \Rightarrow 2x + 6 = 1\)
\(2x = -5 \Rightarrow x = -\frac{5}{2}\)
Step 3: Final Answer
\[ -\frac{5}{2} \]
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