Q.Find the area of the triangle with vertices $A(1, 1, 2)$, $B(2, 3, 5)$ and $C(1, 5, 5)$.

Concept: Cross Product Area — the area of a triangle in 3D is half the magnitude of the cross product of two side vectors.

Step 1: Form side vectors from $A$:

Step 2: Compute the cross product:

$= \mathbf{i}(6 - 12) - \mathbf{j}(3 - 0) + \mathbf{k}(4 - 0) = (-6, -3, 4)$ …

The key insight here is that the cross product of two vectors gives a vector whose magnitude equals the area of the parallelogram they span. A triangle is exactly half of that parallelogram — so the area of triangle $ABC$ is simply $\frac{1}{2} |\vec{AB} \times \vec{AC}|$.

Why does this work? The magnitude $|\vec{u} \times \vec{v}| = |\vec{u}||\vec{v}|\sin\theta$, where $\theta$ is the angle between them. And the area of a triangle with sides $\vec{u}$ and $\vec{v}$ is $\frac{1}{2} |\vec{u}||\vec{v}|\sin\theta$ — exactly the same expression. So the cross product directly encodes the area, no trigonometry needed.

Let's work through it.

1. Choose two side vectors from the same vertex. Pick vertex $A$ as the common starting point. Then:

2. Compute the cross product $\vec{AB} \times \vec{AC}$. Using the determinant formula:

Expand:

$= (-6, -3, 4)$ …

Method: Area of a Triangle from the Cross Product

The magnitude of a cross product is the area of the parallelogram two vectors span; a triangle is half of it.

Steps

Step 1: Form two side vectors from one common vertex.

For vertices $A,B,C$, take $\vec{AB}=B-A$ and $\vec{AC}=C-A$.

Step 2: Cross them and take the magnitude.

$|\vec{AB}\times\vec{AC}| = \text{area of the parallelogram}$ …

Common Mistakes

Mistake 1: Forgetting the factor $\frac12$.

Why it's wrong: the cross-product magnitude gives the parallelogram area; a triangle is half of it. Correct approach: area $=\frac12|\vec{AB}\times\vec{AC}|$.

Mistake 2: Using side vectors from different vertices.

Why it's wrong: both side vectors must start at the same vertex, e.g. $\vec{AB}$ and $\vec{AC}$ from $A$; mixing $\vec{AB}$ with $\vec{BC}$ misdescribes the triangle. Correct approach: take two sides sharing one common vertex. …