Q.Find the angle between the diagonals of a parallelogram having adjacent sides a⃗ = î + 2ĵ + 3k̂ and b⃗ = 2î − 2ĵ + k̂.

Parallelogram Diagonal Vectors – From Scratch

A parallelogram is a slanted rectangle: opposite sides are equal and parallel. Draw both diagonals — each runs from one corner to the opposite corner. The question is: how do we describe these diagonals using the two side vectors that start from the same corner?

The Setup

Take a parallelogram with vertices $A$, $B$, $C$, $D$ in order, with $A$ at the origin. From $A$, two vectors emerge:

• $\vec{a}$ goes from $A$ to $B$ (one side)
• $\vec{b}$ goes from $A$ to $D$ (the other side)

Because opposite sides are equal, $B$ to $C$ is also $\vec{b}$ and $D$ to $C$ is also $\vec{a}$, so the fourth vertex $C$ sits at $\vec{a} + \vec{b}$. The two diagonals run from $A$ to $C$ and from $B$ to $D$.

The Diagonal from the Common Vertex

From $A$ to $C$ you go $\vec{a}$ then $\vec{b}$, ending at the opposite corner:

That's the vector sum of the two sides — walk along one side then the other and you land on the opposite corner.

The Other Diagonal

$B$ is at $\vec{a}$ and $D$ is at $\vec{b}$. To go from $B$ to $D$, you travel from $\vec{a}$ to $\vec{b}$:

The reverse, from $D$ to $B$, is $\vec{a} - \vec{b}$. Both are correct; they just differ in direction.

The Precise Statement

Why This Matters

• Vector addition — the diagonal from the common vertex is the sum of the sides. This is the parallelogram law of vector addition.
• Finding midpoints — the diagonals bisect each other; both midpoints are the same point, $\frac{\vec{a} + \vec{b}}{2}$.
• Physics — the resultant of two forces acting at a point is the diagonal of the parallelogram formed by the force vectors.

A Quick Check

Take $\vec{a} = (3, 0)$ (horizontal) and $\vec{b} = (1, 2)$ (slanted). Then:

• Diagonal from the common vertex: $(3,0) + (1,2) = (4,2)$ …