Q.The vectors $\vec{a}=3\hat{i}-2\hat{j}+2\hat{k}$ and $\vec{b}=-\hat{i}-2\hat{k}$ are the adjacent sides of a parallelogram. The acute angle between its diagonals is ________.

The diagonals of a parallelogram with adjacent sides $\vec{a},\vec{b}$ are $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$.

With $\vec{a}=3\hat{i}-2\hat{j}+2\hat{k}$ and $\vec{b}=-\hat{i}-2\hat{k}$:

Then

$\vec{d}_1\cdot\vec{d}_2=8+4+0=12,\quad |\vec{d}_1|=\sqrt{8}=2\sqrt{2},\quad |\vec{d}_2|=\sqrt{36}=6.$ …

Setup

For a parallelogram with adjacent sides $\vec{a}$ and $\vec{b}$, the two diagonals are the sum and difference of the sides:

Form the diagonals

With $\vec{a}=3\hat{i}-2\hat{j}+2\hat{k}$ and $\vec{b}=-\hat{i}+0\hat{j}-2\hat{k}$:

Dot product and magnitudes

$|\vec{d}_2|=\sqrt{4^2+(-2)^2+4^2}=\sqrt{16+4+16}=\sqrt{36}=6.$ …

Method: Angle Between the Diagonals of a Parallelogram

Use this when the adjacent sides are given and the angle between the diagonals is required.

Steps

Step 1: Form the diagonals from the sides

For sides $\vec{a},\vec{b}$, the diagonals are $\vec{d}_1=\vec{a}+\vec{b}$ and $\vec{d}_2=\vec{a}-\vec{b}$ (the sum and difference of the sides).

Step 2: Apply the dot-product angle formula

Compute the dot product and both magnitudes from components. …

Common Mistakes

Mistake 1: Reporting the obtuse angle instead of the acute one

Why it's wrong: the diagonals make two supplementary angles ($45^\circ$ and $135^\circ$); the question asks for the acute one. Correct approach: if $\cos\theta<0$, take the supplement (or use $|\cos\theta|$) to report the acute value.

Mistake 2: Using the sides (or $\vec{a}\times\vec{b}$) instead of the diagonals

Why it's wrong: the angle is between $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$, not between $\vec{a}$ and $\vec{b}$. Correct approach: form the diagonals first, then apply the dot-product formula. …