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

Concept: Unit Vector Verification — A vector is a unit vector if its magnitude equals 1.

Step 1: Write the given vector:

$\vec{v} = x(\hat{i} + \hat{j} + \hat{k}) = x\hat{i} + x\hat{j} + x\hat{k}$.

Step 2: Compute its magnitude:

$|\vec{v}| = \sqrt{x^2 + x^2 + x^2} = \sqrt{3x^2} = |x|\sqrt{3}$. …

Why This Works: The Unit Vector Idea

A unit vector is any vector whose length (magnitude) is exactly 1. The vector given here is $x$ multiplied by $(\hat{i} + \hat{j} + \hat{k})$. So the question is: what value of $x$ scales this particular direction vector so that its final length becomes 1?

The key insight: multiplying a vector by a scalar $x$ scales its magnitude by $|x|$. If we know the magnitude of $(\hat{i} + \hat{j} + \hat{k})$, we can find the $x$ that makes the product's magnitude equal to 1.

Step-by-Step Solution

1. Find the magnitude of the base vector $\hat{i} + \hat{j} + \hat{k}$.

Each component is 1. So:

2. Understand how scalar multiplication affects magnitude.

If $\vec{v} = \hat{i} + \hat{j} + \hat{k}$, then $x\vec{v}$ has magnitude:

3. Set the magnitude equal to 1 and solve.

We want $|x\vec{v}| = 1$, so: …

Method: Scaling a Vector to Unit Length

Use this whenever a scalar multiple of a fixed vector must be a unit vector, and you must find the scalar.

Steps

Step 1: Use the scaling property of magnitude

For $x\vec v$, the length scales by $|x|$:

The absolute value matters — a negative scalar reverses direction but the length is still positive.

Step 2: Compute $|\vec v|$ and set the product to 1 …

Common Mistakes

Mistake 1: Reporting only the positive value of $x$

Why it's wrong: $|x|=1/\sqrt3$ gives both $x=+\tfrac1{\sqrt3}$ and $x=-\tfrac1{\sqrt3}$; the negative one is an equally valid unit vector pointing the opposite way. Correct approach: keep the $\pm$.

Mistake 2: Forgetting the $\sqrt3$ from the three equal components …