Q.Find the area of the parallelogram whose adjacent sides are determined by the vectors $\vec{a}=\hat i-\hat j+3\hat k$ and $\vec{b}=2\hat i-7\hat j+\hat k$.

The area of a parallelogram formed by two adjacent vectors is the magnitude of their cross product.

Step 1 – Compute the cross product

Step 2 – Simplify each component

$= \hat i[-1 + 21] - \hat j[1 - 6] + \hat k[-7 + 2] = 20\hat i + 5\hat j - 5\hat k$ …

The area of a parallelogram with adjacent sides $\vec a$ and $\vec b$ equals $|\vec a \times \vec b|$.

$\vec a \times \vec b = \begin{vmatrix} \hat i & \hat j & \hat k \\1 & -1 & 3 \\2 & -7 & 1 \end{vmatrix} = \big((-1)(1)-(3)(-7)\big)\hat i - \big((1)(1)-(3)(2)\big)\hat j + \big((1)(-7)-(-1)(2)\big)\hat k$ …

Method: Area of a Parallelogram from the Cross Product

When two vectors are given as the adjacent sides of a parallelogram (or triangle) and you need the area, the tool is the cross product — its magnitude is the area.

Steps

Step 1: Recognise the geometry and pick the right formula

Two vectors $\vec a$ and $\vec b$ from a common vertex span a parallelogram of area

A triangle on the same two sides has half this, $\tfrac12|\vec a\times\vec b|$. Use $\sin\theta$ (not $\cos\theta$) — area is about the perpendicular spread between the vectors.

Step 2: Compute the cross product as a $3\times3$ determinant …

Common Mistakes

Mistake 1: Using the dot product instead of the cross product

Why it's wrong: the dot product gives $|\vec a||\vec b|\cos\theta$, which measures alignment, not area. Correct approach: area needs the cross product magnitude $|\vec a\times\vec b|=|\vec a||\vec b|\sin\theta$.

Mistake 2: Dropping the minus sign on the $\hat j$ component …