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

(B) $\frac{\vec{b}+2 \vec{d}}{3}$ is the position vector of a point which divides PR internally in the ratio 5 : 4

(A) The points $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ and S are NOT coplanar

(C) $\frac{\overrightarrow{\mathrm{b}}+2 \overrightarrow{\mathrm{~d}}}{3}$ is the position vector of a point which divides PR externally in the ratio 5 : 4

(D) The square of the magnitude of the vector $\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{d}}$ is 95

JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Vector Algebra

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Let the position vectors of the points

P, Q, R

and

S

a = \hat{i} + 2 \hat{j} - 5 \hat{k}, b = 3 \hat{i} + 6 \hat{j} + 3 \hat{k}

c = \frac{17}{5} \hat{i} + \frac{16}{5} \hat{j} + 7 \hat{k}

and

d = 2 \hat{i} + \hat{j} + \hat{k}

, respectively. Then which of the following statements is true?

(A)The points $P, Q, R$ and S are NOT coplanar
(B) $\frac{b + 2 d}{3}$ is the position vector of a point which divides PR internally in the ratio 5 : 4✓ Correct
(C) $\frac{b + 2 d}{3}$ is the position vector of a point which divides PR externally in the ratio 5 : 4
(D)The square of the magnitude of the vector $b \times d$ is 95

Answer: (B)

Solution

Calculate

\frac{b + 2 d}{3} = \frac{( 3 i ^ + 6 j ^ + 3 k ^ ) + 2 ( 2 i ^ + j ^ + k ^ )}{3} = \frac{7 i ^ + 8 j ^ + 5 k ^}{3}

For internal division of PR in ratio 5:4:

\frac{5 c + 4 a}{9} = \frac{5 ( \frac{17}{5} i ^ + \frac{16}{5} j ^ + 7 k ^ ) + 4 ( i ^ + 2 j ^ - 5 k ^ )}{9}

= \frac{17 i ^ + 16 j ^ + 35 k ^ + 4 i ^ + 8 j ^ - 20 k ^}{9} = \frac{21 i ^ + 24 j ^ + 15 k ^}{9} = \frac{7 i ^ + 8 j ^ + 5 k ^}{3}

This matches

\frac{b + 2 d}{3}

, so option (B) is correct.

For option (D):

∣ b \times d ∣^{2} = ∣ b ∣^{2} ∣ d ∣^{2} - (b \cdot d)^{2}

= (9 + 36 + 9) (4 + 1 + 1) - (6 + 6 + 3)^{2} = 54 \times 6 - 225 = 324 - 225 = 99 \neq = 95

JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Single correct option

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JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Vector Algebra

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