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

• A. \(\frac{-5}{2}\)
• B. 6
• C. -6
• D. 3

Hint:

Scalar projection = \(\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}\).

Answer 1

The Correct Option is A

To find the value of \(x\) in the given vector equation, we need to use the concept of the scalar projection of one vector onto another.

The scalar projection of a vector \(\mathbf{a}\) onto another vector \(\mathbf{b}\) is given by the formula:

\[\text{Scalar Projection of } \mathbf{a} \text{ on } \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|}\]

Here, we have:

• Vector \(\mathbf{a} = x\mathbf{i} - \mathbf{j} + \mathbf{k}\)
• Vector \(\mathbf{b} = 2\mathbf{i} - \mathbf{j} + 5\mathbf{k}\)

Given that this scalar projection is \(\frac{1}{\sqrt{30}}\), we need to calculate the following:

1. Compute the dot product \(\mathbf{a} \cdot \mathbf{b}\):

\(\mathbf{a} \cdot \mathbf{b} = (x\cdot 2) + (-1)\cdot (-1) + (1)\cdot(5) = 2x + 1 + 5 = 2x + 6\)

1. Compute the magnitude of vector \(\mathbf{b}\), \(\|\mathbf{b}\|\):

\(\|\mathbf{b}\| = \sqrt{2^2 + (-1)^2 + 5^2} = \sqrt{4 + 1 + 25} = \sqrt{30}\)

The scalar projection equation now becomes:

\(\frac{2x + 6}{\sqrt{30}} = \frac{1}{\sqrt{30}}\)

By multiplying both sides by \(\sqrt{30}\), we obtain:

\((2x + 6) = 1\)

Simplifying this equation, we find:

\(2x = 1 - 6\)

\(2x = -5\)

\(x = \frac{-5}{2}\)

Therefore, the value of \(x\) is \(\frac{-5}{2}\) which is the correct answer.