Q.Show that the points A, B and C with position vectors, $\vec{a} = 3\hat{i} - 4\hat{j} - 4\hat{k}$, $\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{c} = \hat{i} - 3\hat{j} - 5\hat{k}$, respectively form the vertices of a right angled triangle.

Concept: Collinearity Condition & Right-Angle Test

We check if the vectors representing two sides are perpendicular (dot product = 0).

Step 1 – Find side vectors

$\overrightarrow{AB} = \vec{b} - \vec{a} = (2-3)\hat{i} + (-1+4)\hat{j} + (1+4)\hat{k} = -\hat{i} + 3\hat{j} + 5\hat{k}$

$\overrightarrow{AC} = \vec{c} - \vec{a} = (1-3)\hat{i} + (-3+4)\hat{j} + (-5+4)\hat{k} = -2\hat{i} + \hat{j} - \hat{k}$

Step 2 – Compute dot product

$\overrightarrow{AB} \cdot \overrightarrow{AC} = (-1)(-2) + (3)(1) + (5)(-1) = 2 + 3 - 5 = 0$ …

We are given three points in space. To show they form a right-angled triangle, we don’t need to find all three angles — just one pair of perpendicular sides is enough. The condition for perpendicularity is that the dot product of the corresponding side vectors is zero.

Why this works:

If three points are non-collinear, they always form a triangle. A right-angled triangle is simply a triangle where one interior angle is $90^\circ$. That angle is formed by two sides meeting at a vertex. So we pick a vertex, form the two vectors that start at that vertex and go to the other two points, and check if their dot product is zero.

Let’s label the points:

• $A$: $\vec{a} = 3\hat{i} - 4\hat{j} - 4\hat{k}$
• $B$: $\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$
• $C$: $\vec{c} = \hat{i} - 3\hat{j} - 5\hat{k}$

We’ll compute the side vectors and test each vertex.

1. Find $\overrightarrow{AB}$ and $\overrightarrow{AC}$ (vertex A).

   $\overrightarrow{AB} = \vec{b} - \vec{a} = (2-3)\hat{i} + (-1+4)\hat{j} + (1+4)\hat{k} = -\hat{i} + 3\hat{j} + 5\hat{k}$

   $\overrightarrow{AC} = \vec{c} - \vec{a} = (1-3)\hat{i} + (-3+4)\hat{j} + (-5+4)\hat{k} = -2\hat{i} + \hat{j} - \hat{k}$

   Dot product:

   $\overrightarrow{AB} \cdot \overrightarrow{AC} = (-1)(-2) + (3)(1) + (5)(-1) = 2 + 3 - 5 = 0$

   Since the dot product is zero, $\overrightarrow{AB} \perp \overrightarrow{AC}$. That means angle at A is $90^\circ$.

2. Verify that the points are not collinear (so they truly form a triangle).

   If they were collinear, all side vectors would be parallel. Here $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are perpendicular, so they are definitely not parallel. Hence A, B, C are non-collinear.

3. Optional: Check the other two vertices for completeness. …

Method: Proving Three Points Form a Right-Angled Triangle

Use this whenever you must show that points $A,B,C$ (given by position vectors) form a right-angled triangle. The strategy: find the side vectors from a common vertex and test for a perpendicular pair.

Steps

Step 1: Build side vectors from a chosen vertex.

At vertex $A$: $\vec{AB}=\vec{b}-\vec{a}$ and $\vec{AC}=\vec{c}-\vec{a}$. Always subtract position vectors to get the side vectors — never dot the position vectors themselves.

Step 2: Test perpendicularity with the dot product.

$\vec{AB}\cdot\vec{AC}=0\iff \text{right angle at }A$ …

Common Mistakes

Mistake 1: Dotting the position vectors instead of the side vectors.

Why it's wrong: $\vec{a}\cdot\vec{b}$ measures directions from the origin, telling you nothing about the triangle's interior angle. Correct approach: form side vectors $\vec{AB}=\vec{b}-\vec{a}$, $\vec{AC}=\vec{c}-\vec{a}$ first, then take their dot product.

Mistake 2: Testing only one vertex and stopping.

Why it's wrong: if $\vec{AB}\cdot\vec{AC}\neq0$, the right angle may still be at $B$ or $C$. Correct approach: try other vertex pairs before concluding the triangle is not right-angled. …