Let the position vectors of the points P, Q, R and S be ā = î + 2ĵ - 5k̂, b̄ = 3î + 6ĵ + 3k̂, c̄ = 17/5î + 16/5ĵ + 7k̂ and d̄ = 2î + ĵ + k̂, respectively. Then which of the following statements is true?

Let the position vectors of the points P, Q, R and S be ā = î + 2ĵ - 5k̂, b̄ = 3î + 6ĵ + 3k̂, c̄ = 17/5î + 16/5ĵ + 7k̂ and d̄ = 2î + ĵ + k̂, respectively. Then which of the following statements is true? Given: a⃗ = î + 2ĵ - 5k̂ b⃗ = 3î + 6ĵ + 3k̂, c⃗ = 17/5î + 16/5ĵ + 7k̂ d⃗ = 2î + ĵ + k̂ To find: which of the four statements is true? (1) not coplanar (2) internal 5 : 4 (3) external 5 : 4 (4) |b⃗ × d⃗|² = 95 Plan: (1) coplanarity: (b⃗-a⃗)·[(c⃗-a⃗)×(d⃗-a⃗)] = 0 ? (2),(3): section formula on PR, internal vs external 5 : 4 (4): b⃗×d⃗, then |b⃗×d⃗|² PQ⃗ = b⃗-a⃗ = (2, 4, 8) PR⃗ = c⃗-a⃗ = (12/5, 6/5, 12) PS⃗ = d⃗-a⃗ = (1, -1, 6) PR⃗×PS⃗ = (96/5, -12/5, -18/5) PQ⃗·(PR⃗×PS⃗) = 192/5-48/5-144/5 = 0 ⇒ P,Q,R,S are coplanar ⇒ (1) is false (b⃗+2d⃗)/3 = ((3,6,3)+2(2,1,1))/3 = (7,8,5)/3 = (7/3, 8/3, 5/3) T = (4a⃗+5c⃗)/9 = ((4,8,-20)+(17,16,35))/9 = (21,24,15)/9 T = (7/3, 8/3, 5/3) T' = (5c⃗-4a⃗)/(5-4) = (13, 8, 55) ≠ (7/3, 8/3, 5/3) ⇒ (2) is true, (3) is false b⃗×d⃗ = î ĵ k̂ 3 6 3 2 1 1 = î(6-3) - ĵ(3-6) + k̂(3-12) = 3î + 3ĵ - 9k̂ |b⃗×d⃗|² = 9+9+81 = 99 ≠ 95 ⇒ (4) is false Answer: (2) (b⃗+2d⃗)/3 = (7/3,8/3,5/3) divides PR internally in 5:4

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

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

Vector Algebra Let the position vectors of the points P, Q, R and S be

\bar{a} = \hat{i} + 2\hat{j} - 5\hat{k}

\bar{b} = 3\hat{i} + 6\hat{j} + 3\hat{k}

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

and

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

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

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Written solution

$\text{Given: } \vec{a} = \hat{i} + 2\hat{j} - 5\hat{k}$ $\vec{b} = 3\hat{i} + 6\hat{j} + 3\hat{k}, \quad \vec{c} = \dfrac{17}{5}\hat{i} + \dfrac{16}{5}\hat{j} + 7\hat{k}$ $\vec{d} = 2\hat{i} + \hat{j} + \hat{k}$ $\text{To find: which of the four statements is true?}$ $(\text{1})\ \text{not coplanar} \qquad (\text{2})\ \text{internal } 5 : 4$ $(\text{3})\ \text{external } 5 : 4 \qquad (\text{4})\ |\vec{b} \times \vec{d}|^2 = 95$ $\text{Plan: (1) coplanarity: } (\vec{b}-\vec{a})\cdot[(\vec{c}-\vec{a})\times(\vec{d}-\vec{a})] = 0\,?$ $(2),(3): \text{ section formula on } PR, \text{ internal vs external } 5 : 4$ $(4): \ \vec{b}\times\vec{d}, \text{ then } |\vec{b}\times\vec{d}|^2$ $\vec{PQ} = \vec{b}-\vec{a} = (2,\ 4,\ 8)$ $\vec{PR} = \vec{c}-\vec{a} = \left(\dfrac{12}{5},\ \dfrac{6}{5},\ 12\right)$ $\vec{PS} = \vec{d}-\vec{a} = (1,\ -1,\ 6)$ $\vec{PR}\times\vec{PS} = \left(\dfrac{96}{5},\ -\dfrac{12}{5},\ -\dfrac{18}{5}\right)$ $\vec{PQ}\cdot(\vec{PR}\times\vec{PS}) = \dfrac{192}{5}-\dfrac{48}{5}-\dfrac{144}{5} = 0$ $\Rightarrow P,Q,R,S \text{ are coplanar} \Rightarrow (1) \text{ is false}$ $\dfrac{\vec{b}+2\vec{d}}{3} = \dfrac{(3,6,3)+2(2,1,1)}{3}$ $= \dfrac{(7,8,5)}{3} = \left(\dfrac{7}{3},\ \dfrac{8}{3},\ \dfrac{5}{3}\right)$ $T = \dfrac{4\vec{a}+5\vec{c}}{9}$ $= \dfrac{(4,8,-20)+(17,16,35)}{9} = \dfrac{(21,24,15)}{9}$ $T = \left(\dfrac{7}{3},\ \dfrac{8}{3},\ \dfrac{5}{3}\right)$ $T' = \dfrac{5\vec{c}-4\vec{a}}{5-4} = (13,\ 8,\ 55) \neq \left(\dfrac{7}{3},\ \dfrac{8}{3},\ \dfrac{5}{3}\right)$ $\Rightarrow (2) \text{ is true, } (3) \text{ is false}$ $\vec{b}\times\vec{d} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 6 & 3 \\ 2 & 1 & 1 \end{vmatrix}$ $= \hat{i}(6-3) - \hat{j}(3-6) + \hat{k}(3-12) = 3\hat{i} + 3\hat{j} - 9\hat{k}$ $|\vec{b}\times\vec{d}|^2 = 9+9+81 = 99 \neq 95$ $\Rightarrow (4) \text{ is false}$ $\text{Answer: } (2)$ $\dfrac{\vec{b}+2\vec{d}}{3} = \left(\dfrac{7}{3},\dfrac{8}{3},\dfrac{5}{3}\right) \text{ divides } PR \text{ internally in } 5:4$

Written solution

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