Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade 12
Question:
Let the position vectors of the points A, B, C be $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = -\hat{i} - \hat{j} + 8\hat{k}$ and $\vec{c} = -4\hat{i} + 4\hat{j} + 6\hat{k}$ respectively, then:
$\triangle ABC$ is equilateral
$\triangle ABC$ is right angled
$|\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 = 2|\vec{a} - \vec{b}|^2$
$|\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 = |\vec{a} - \vec{b}|^2$
Step-by-Step Solution
Key Concept: An equilateral triangle has all sides equal, which automatically satisfies the relation $|\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 = 2|\vec{a} - \vec{b}|^2$ since each side has the same length.
First, compute the side vectors: $\vec{AB} = \vec{b} - \vec{a} = -2\hat{i} - 3\hat{j} + 5\hat{k}$, $\vec{BC} = \vec{c} - \vec{b} = -3\hat{i} + 5\hat{j} - 2\hat{k}$, $\vec{CA} = \vec{a} - \vec{c} = 5\hat{i} - 2\hat{j} - 3\hat{k}$. Calculate the squared magnitudes: $|\vec{AB}|^2 = 4 + 9 + 25 = 38$, $|\vec{BC}|^2 = 9 + 25 + 4 = 38$, $|\vec{CA}|^2 = 25 + 4 + 9 = 38$. Since all three sides are equal, $\triangle ABC$ is equilateral (Option 1 is correct). For Option 3, compute: $|\vec{BC}|^2 + |\vec{CA}|^2 = 38 + 38 = 76$ and $2|\vec{AB}|^2 = 2(38) = 76$, confirming the relation holds (Option 3 is correct).
Correct Answer: 1,3