JEE Main 2026 — 24 January, Morning Shift — Question 21

$\mathrm{d}_{1}=<4,1,1>$ and $\mathrm{d}_{2}=<1,1,0>$ $d_{L}=d_{1} \times d_{2}=\left|\begin{array}{lll} i & j & k \\4 & 1 & 1 \\1 & 1 & 0 \end{array}\right|=<-1,1,3>$ Line L passes through $\mathrm{P}<1,1,1>$ with $\mathrm{d}_{2}=<-1,1,3>$

$$\begin{aligned} & \mathrm{r}(\mathrm{t})=<1,1,1>+\mathrm{t}<-1,1,3> & =<1-\mathrm{t}, 1+\mathrm{t}, 1+3 \mathrm{t}> & =<1-\mathrm{t}, 1+\mathrm{t}, 1+3 \mathrm{t}> \end{aligned}$$

For point $\mathrm{Q} ; \mathrm{x}=0$

$\Rightarrow \mathrm{t}=1$ $\mathrm{Q}=<0,2,4>$

Another Line parallel to L passes through $\mathrm{S}<1,0,-1>$ with $\mathrm{d}_{\mathrm{L}}=\langle-1,1,3\rangle$

$\left.\mathrm{r}^{\prime}(\mathrm{u})=\langle 1,0,-1\rangle+\mu<-1,1,3\right\rangle$

$=<1-\mu, \mu,-1+3 \mu>$ for point $\mathrm{R}, \mathrm{x}=0$

$\Rightarrow \mu=1$ $\mathrm{R}<0,1,2>$

Area of parallelogram with adjacent vectors $\overrightarrow{\mathrm{PQ}}$ and $\overrightarrow{\mathrm{PS}}$ $\overrightarrow{\mathrm{PQ}}=\langle-1,1,3\rangle$ $\overrightarrow{\mathrm{PS}}=\langle 0,-1,-2\rangle$

Area of parallelogram $\begin{aligned} & \overrightarrow{\mathrm{PQ}} \times \overrightarrow{\mathrm{PS}}=\left|\begin{array}{ccc} \mathrm{i} & \mathrm{j} & \mathrm{k} \\-1 & 1 & 3 \\0 & -1 & -2 \end{array}\right|=<1,-2,1> \end{aligned}$

$${ Area }=\sqrt{1^{2}+(-2)^{2}+1^{2}}=\sqrt{6}$$

the square of the area of the parallelogram PQRS is equal to $6$