Q.Find the sum of the vectors $\vec{a} = \hat{i} - 2\hat{j} + \hat{k}$, $\vec{b} = -2\hat{i} + 4\hat{j} + 5\hat{k}$ and $\vec{c} = \hat{i} - 6\hat{j} - 7\hat{k}$.

Concept: Vector Addition — add corresponding components.

Step 1: Write the vectors in component form:

$\vec{a} = (1, -2, 1)$, $\vec{b} = (-2, 4, 5)$, $\vec{c} = (1, -6, -7)$.

Step 2: Add the $\hat{i}$ components: $1 + (-2) + 1 = 0$.

Step 3: Add the $\hat{j}$ components: $-2 + 4 + (-6) = -4$. …

The idea is simple: when you add vectors, you combine their effects along each direction independently. Think of it like adding apples to apples and oranges to oranges — the $\hat{i}$ parts only combine with other $\hat{i}$ parts, and so on. This works because the unit vectors $\hat{i}, \hat{j}, \hat{k}$ are mutually perpendicular and form a basis for 3D space.

Let’s go through it step by step.

1. Identify the components of each vector.

   $\vec{a} = 1\hat{i} - 2\hat{j} + 1\hat{k}$

   $\vec{b} = -2\hat{i} + 4\hat{j} + 5\hat{k}$

   $\vec{c} = 1\hat{i} - 6\hat{j} - 7\hat{k}$

2. Add the $\hat{i}$-components.

   $1 + (-2) + 1 = 0$

   So the $\hat{i}$-component of the sum is $0\hat{i}$ — it cancels out completely.

3. Add the $\hat{j}$-components.

   $-2 + 4 + (-6) = -4$

   So the $\hat{j}$-component is $-4\hat{j}$.

4. Add the $\hat{k}$-components.

   $1 + 5 + (-7) = -1$

   So the $\hat{k}$-component is $-\hat{k}$.

5. Write the resultant vector. …

Method: Adding vectors component-wise

Use this whenever you must sum two or more vectors given in $\hat{i},\hat{j},\hat{k}$ form.

Steps

Step 1: Line up like components

Because $\hat{i},\hat{j},\hat{k}$ are mutually perpendicular basis directions, each combines only with its own kind — "$\hat i$ with $\hat i$", and so on.

Step 2: Add each component separately

$\sum\vec{v}=\Big(\textstyle\sum x\Big)\hat{i}+\Big(\textstyle\sum y\Big)\hat{j}+\Big(\textstyle\sum z\Big)\hat{k}$ …

Common Mistakes

Mistake 1: Sign errors when adding negative components

Why it's wrong: the $\hat j$ sum is $-2+4-6=-4$ and the $\hat k$ sum is $1+5-7=-1$; a dropped minus gives a wrong resultant. Correct approach: add each column carefully with signs, giving $0\hat i-4\hat j-\hat k$.

Mistake 2: Mixing components across directions …