JEE Advanced 2023 Paper 2 Q04 Mathematics Vectors & 3D Geometry Vector Algebra Medium

JEE Advanced 2023 Paper 2 · Q04 · Vector Algebra

Let the position vectors of the points $P$, $Q$, $R$ and $S$ be $\vec{a} = \hat{i} + 2\hat{j} - 5\hat{k}$, $\vec{b} = 3\hat{i} + 6\hat{j} + 3\hat{k}$, $\vec{c} = \dfrac{17}{5}\hat{i} + \dfrac{16}{5}\hat{j} + 7\hat{k}$ and $\vec{d} = 2\hat{i} + \hat{j} + \hat{k}$, respectively. Then which of the following statements is true?

  1. A. The points $P$, $Q$, $R$ and $S$ are NOT coplanar
  2. B. $\dfrac{\vec{b} + 2\vec{d}}{3}$ is the position vector of a point which divides $PR$ internally in the ratio $5 : 4$
  3. C. $\dfrac{\vec{b} + 2\vec{d}}{3}$ is the position vector of a point which divides $PR$ externally in the ratio $5 : 4$
  4. D. The square of the magnitude of the vector $\vec{b} + \vec{d}$ is $95$
Reveal answer + step-by-step solution

Correct answer:B

Solution

$\overrightarrow{PR} = \vec{c} - \vec{a} = (12/5, 6/5, 12)$, $\overrightarrow{PQ} = (2, 4, 8)$, $\overrightarrow{PS} = (1, -1, 6)$. Box product = 0 (coplanar) — (A) FALSE. Compute $(\vec{b}+2\vec{d})/3 = (7, 8, 5)/3$. Internal division of $PR$ in $5:4$: $\dfrac{5\vec{c}+4\vec{a}}{9} = \dfrac{(17+4, 16+8, 35-20)}{9} = \dfrac{(21,24,15)}{9} = (7,8,5)/3$ ✓ — (B) TRUE. (C) is the same value with wrong external interpretation — FALSE. $|\vec{b}+\vec{d}|^2 = (5)^2 + (7)^2 + (4)^2 = 25+49+16 = 90 \ne 95$ — (D) FALSE.

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