Q31 / 56 · IAT (IISER) 2026 · Mathematics
Consider the points $A(4\hat{i} + \hat{j} + 3\hat{k})$, $B(2\hat{j})$ and $C(-4\hat{i} + 3\hat{j} - 3\hat{k})$. Which of the following statements is TRUE?
- A$\overrightarrow{AB} \times \overrightarrow{BC} = \hat{i} + \hat{j} + \hat{k}$.
- B$\overrightarrow{AB} + 3\overrightarrow{BC}$ is perpendicular to $\overrightarrow{AC}$.
- C$\overrightarrow{AB}$, $\overrightarrow{BC}$ and $\overrightarrow{CA}$ are mutually perpendicular.
- Correct answerD$A$, $B$ and $C$ are collinear.
Solution
Position vectors of the points are given as $\vec{A} = 4\hat{i} + \hat{j} + 3\hat{k}$, $\vec{B} = 2\hat{j}$, and $\vec{C} = -4\hat{i} + 3\hat{j} - 3\hat{k}$.
The vectors $\overrightarrow{AB}$ and $\overrightarrow{BC}$ are calculated as follows:
$\overrightarrow{AB} = \vec{B} - \vec{A} = (0 - 4)\hat{i} + (2 - 1)\hat{j} + (0 - 3)\hat{k} = -4\hat{i} + \hat{j} - 3\hat{k}$
$\overrightarrow{BC} = \vec{C} - \vec{B} = (-4 - 0)\hat{i} + (3 - 2)\hat{j} + (-3 - 0)\hat{k} = -4\hat{i} + \hat{j} - 3\hat{k}$
Since $\overrightarrow{AB} = \overrightarrow{BC}$, the vectors are parallel and share the common point $B$.
Therefore, the points $A$, $B$, and $C$ are collinear.
Answer: $A$, $B$ and $C$ are collinear.
The vectors $\overrightarrow{AB}$ and $\overrightarrow{BC}$ are calculated as follows:
$\overrightarrow{AB} = \vec{B} - \vec{A} = (0 - 4)\hat{i} + (2 - 1)\hat{j} + (0 - 3)\hat{k} = -4\hat{i} + \hat{j} - 3\hat{k}$
$\overrightarrow{BC} = \vec{C} - \vec{B} = (-4 - 0)\hat{i} + (3 - 2)\hat{j} + (-3 - 0)\hat{k} = -4\hat{i} + \hat{j} - 3\hat{k}$
Since $\overrightarrow{AB} = \overrightarrow{BC}$, the vectors are parallel and share the common point $B$.
Therefore, the points $A$, $B$, and $C$ are collinear.
Answer: $A$, $B$ and $C$ are collinear.
Hence, the correct answer is option: 'D'.