Let the position vectors of points (P, Q, R) and (S) be as given below. Which statement is true?

JEE Advanced medium Year: 2023 Maths
Let the position vectors of points \(P, Q, R\) and \(S\) be

\[ \vec a = \hat i + 2\hat j - 5\hat k,\quad \vec b = 3\hat i + 6\hat j + 3\hat k,\quad \vec c = \frac{17}{5}\hat i + \frac{16}{5}\hat j + 7\hat k,\quad \vec d = 2\hat i + \hat j + \hat k. \]

Then which of the following statements is true?

(A) The points \(P, Q, R, S\) are not coplanar.

(B) \(\displaystyle \frac{\vec b + 2\vec d}{3}\) divides \(PR\) internally in the ratio \(5:4\).

(C) \(\displaystyle \frac{\vec b + 2\vec d}{3}\) divides \(PR\) ernally in the ratio \(5:4\).

(D) The square of magnitude of \(\vec b \times \vec d\) is \(95\).
Answer
B
Solution
Coplanarity.

\[ \vec{PQ}=\vec{b}-\vec{a}=\langle 2,4,8 angle,\quad \vec{PR}=\vec{c}-\vec{a}=leftlangle \frac{12}{5}, \frac{6}{5},12 ight angle,\quad \vec{PS}=\vec{d}-\vec{a}=\langle 1,-1,6 angle. \]
\[ [\vec{PQ},\vec{PR},\vec{PS}] = 0 ;\Rightarrow; \mathrm{coplanar} ;\Rightarrow; \mathrm{(A) false}. \]
Division.

\[ \frac{\vec{b}+2\vec{d}}{3}=\frac{(3,6,3)+2(2,1,1)}{3} =\Big( \frac{7}{3}, \frac{8}{3}, \frac{5}{3}\Big). \]
\[ \mathrm{Point dividing } PR \mathrm{ internally in } 5{:}4 \mathrm{ from } P:; \vec{X}=\frac{5\vec{r}+4\vec{p}}{9}=\frac{5\vec{c}+4\vec{a}}{9} =\Big( \frac{7}{3}, \frac{8}{3}, \frac{5}{3}\Big). \]
Hence (B) true and (C) false.

Cross product.

\[ \vec{b} imes\vec{d}= egin{vmatrix} \hat{i}&\hat{j}&\hat{k}\ 3&6&3\ 2&1&1 \end{vmatrix}=(3,3,-9),\qquad lVert \vec{b} imes\vec{d} Vert^2=3^2+3^2+(-9)^2=99 eq95. \]
Therefore only (B) is correct.