Q.For given vectors, $\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}$ and $\vec{b} = -\hat{i} + \hat{j} - \hat{k}$, find the unit vector in the direction of the vector $\vec{a} + \vec{b}$.

First, find the sum of the two vectors:

Combine like terms:

So the resultant vector is $\hat{i} + \hat{k}$. Its magnitude is:

$|\vec{a} + \vec{b}| = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{2}$ …

Why this approach works

A unit vector in a given direction is simply a vector of length 1 that points the same way. To get it, you take any vector pointing in that direction and scale it down to length 1 — which means dividing the vector by its own magnitude. So the problem breaks into two clean steps: first find $\vec{a}+\vec{b}$, then compute its magnitude and divide.

Step-by-step solution

1. Add the vectors component-wise

Vector addition is straightforward: add the $\hat{i}$ coefficients together, the $\hat{j}$ coefficients together, and the $\hat{k}$ coefficients together.

Adding:

• $\hat{i}$ component: $2 + (-1) = 1$
• $\hat{j}$ component: $-1 + 1 = 0$
• $\hat{k}$ component: $2 + (-1) = 1$

So:

2. Find the magnitude of $\vec{a}+\vec{b}$

The magnitude of a vector $x\hat{i} + y\hat{j} + z\hat{k}$ is $\sqrt{x^2 + y^2 + z^2}$.

For $\hat{i} + \hat{k}$, we have $x=1$, $y=0$, $z=1$:

3. Divide the vector by its magnitude to get the unit vector …

Method: Unit Vector of a Combined Vector

Use this when the direction is given as a combination such as $\vec{a}+\vec{b}$ (or $\vec{a}-\vec{b}$): resolve the combination into a single vector first, then normalise.

Steps

Step 1: Combine the vectors component-wise.

Add (or subtract) the $\hat{i}$ coefficients together, then the $\hat{j}$, then the $\hat{k}$:

Keep signs strict — a component may cancel to $0$, and a zero component is still part of the vector.

Step 2: Find the magnitude of the resulting vector. …

Common Mistakes

Mistake 1: Normalising $\vec{a}$ and $\vec{b}$ separately, then adding.

Why it's wrong: the unit vector of a sum is not the sum of the unit vectors — direction depends on the combined vector. Correct approach: add first to get $\vec{a}+\vec{b}$, then normalise that single vector.

Mistake 2: Sign slips when adding components.

Why it's wrong: here $-\hat{j}+\hat{j}=0$ and $2\hat{k}+(-\hat{k})=\hat{k}$; mishandling a sign changes the whole answer. Correct approach: line up components and add signed numbers carefully.

Mistake 3: Dropping the zero component and mis-computing the magnitude. …