Question

Find the value of \( x \) for which \( x ( \hat{i} + \hat{j} + \hat{k} ) \) is a unit vector.

Hint:

A unit vector has a magnitude of 1. To find \( x \) for a vector \( x \mathbf{v} \) to be a unit vector, set its magnitude equal to 1 and solve for \( x \).

Answer 1

Step 1: Understand the concept of a unit vector.
A unit vector has a magnitude of 1. Therefore, we need to find the value of \( x \) such that the magnitude of the vector \( x ( \hat{i} + \hat{j} + \hat{k} ) \) is equal to 1.

Step 2: Find the magnitude of the vector.
The magnitude of the vector \( \hat{i} + \hat{j} + \hat{k} \) is: \[ \| \hat{i} + \hat{j} + \hat{k} \| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}. \] Now, the magnitude of \( x( \hat{i} + \hat{j} + \hat{k} ) \) is: \[ \| x( \hat{i} + \hat{j} + \hat{k} ) \| = |x| \cdot \| \hat{i} + \hat{j} + \hat{k} \| = |x| \cdot \sqrt{3}. \]

Step 3: Set the magnitude equal to 1.
For the vector to be a unit vector, we set the magnitude equal to 1: \[ |x| \cdot \sqrt{3} = 1. \] Solving for \( x \): \[ |x| = \frac{1}{\sqrt{3}}. \] Thus, the value of \( x \) is: \[ x = \pm \frac{1}{\sqrt{3}}. \]

Step 4: Conclusion.
Therefore, the value of \( x \) for which the vector \( x ( \hat{i} + \hat{j} + \hat{k} ) \) is a unit vector is \( x = \pm \frac{1}{\sqrt{3}} \).