Q.Vectors $\vec{a} = 3\hat{i} - 2\hat{j} + 2\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{k}$ represent the adjacent sides of a parallelogram. Find the vectors representing the diagonals of the parallelogram. Also, find their lengths.

Concept: Parallelogram Diagonal Vectors

If $\vec{a}$ and $\vec{b}$ are adjacent sides, the diagonals are $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ (or $\vec{b} - \vec{a}$; both have the same length).

Step 1 – First diagonal

$\vec{d}_1 = \vec{a} + \vec{b} = (3+1)\hat{i} + (-2+0)\hat{j} + (2+2)\hat{k} = 4\hat{i} - 2\hat{j} + 4\hat{k}$.

Step 2 – Second diagonal

$\vec{d}_2 = \vec{a} - \vec{b} = (3-1)\hat{i} + (-2-0)\hat{j} + (2-2)\hat{k} = 2\hat{i} - 2\hat{j} + 0\hat{k}$.

Step 3 – Lengths …

Why the diagonal vectors are $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$

Imagine a parallelogram with adjacent sides $\vec{a}$ and $\vec{b}$ meeting at a common vertex. The diagonal that goes from that common vertex to the opposite vertex is simply the vector sum $\vec{a} + \vec{b}$ — you travel along $\vec{a}$ then along $\vec{b}$, or vice versa.

The other diagonal connects the tips of $\vec{a}$ and $\vec{b}$ (when placed tail-to-tail). To go from the tip of $\vec{a}$ to the tip of $\vec{b}$, you go backwards along $\vec{a}$ and then forward along $\vec{b}$, giving $\vec{b} - \vec{a}$. Equivalently, starting from the tip of $\vec{b}$ to the tip of $\vec{a}$ gives $\vec{a} - \vec{b}$. Both are valid diagonal vectors — they differ only in direction, and their lengths are the same.

Step-by-step computation

1. Find $\vec{d}_1 = \vec{a} + \vec{b}$

Add component-wise:

• $\hat{i}$: $3 + 1 = 4$
• $\hat{j}$: $-2 + 0 = -2$
• $\hat{k}$: $2 + 2 = 4$

So $\vec{d}_1 = 4\hat{i} - 2\hat{j} + 4\hat{k}$.

2. Find $\vec{d}_2 = \vec{a} - \vec{b}$

Subtract component-wise:

• $\hat{i}$: $3 - 1 = 2$
• $\hat{j}$: $-2 - 0 = -2$
• $\hat{k}$: $2 - 2 = 0$

So $\vec{d}_2 = 2\hat{i} - 2\hat{j} + 0\hat{k} = 2\hat{i} - 2\hat{j}$. …