For what value of lambda is the volume of the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the vertices be $A(\hat{i} - 6\hat{j} + 10\hat{k})$,$B(-\hat{i} - 3\hat{j} + 7\hat{k})$,$C(5\hat{i} - \hat{j} + \lambda\hat{k})$,and $D(7\hat{

Vedclass · Accepted Answer

$1, 7$

Vedclass · Answer

(B) Let the vertices be $A(\hat{i} - 6\hat{j} + 10\hat{k})$,$B(-\hat{i} - 3\hat{j} + 7\hat{k})$,$C(5\hat{i} - \hat{j} + \lambda\hat{k})$,and $D(7\hat{i} - 4\hat{j} + 7\hat{k})$.The vectors representing the edges from vertex $A$ are:$\vec{AB} = B - A = (-1-1)\hat{i} + (-3+6)\hat{j} + (7-10)\hat{k} = -2\hat{i} + 3\hat{j} - 3\hat{k}$$\vec{AC} = C - A = (5-1)\hat{i} + (-1+6)\hat{j} + (\lambda-10)\hat{k} = 4\hat{i} + 5\hat{j} + (\lambda-10)\hat{k}$$\vec{AD} = D - A = (7-1)\hat{i} + (-4+6)\hat{j} + (7-10)\hat{k} = 6\hat{i} + 2\hat{j} - 3\hat{k}$The volume of a tetrahedron is given by $V = \frac{1}{6} |[\vec{AB} \vec{AC} \vec{AD}]| = 11$.So,$|[\vec{AB} \vec{AC} \vec{AD}]| = 66$,which means $[\vec{AB} \vec{AC} \vec{AD}] = \pm 66$.The scalar triple product is the determinant:$\begin{vmatrix} -2 & 3 & -3 \ 4 & 5 & \lambda-10 \ 6 & 2 & -3 \end{vmatrix} = \pm 66$Expanding the determinant:$-2[5(-3) - 2(\lambda-10)] - 3[4(-3) - 6(\lambda-10)] - 3[4(2) - 6(5)] = \pm 66$$-2[-15 - 2\lambda + 20] - 3[-12 - 6\lambda + 60] - 3[8 - 30] = \pm 66$$-2[5 - 2\lambda] - 3[48 - 6\lambda] - 3[-22] = \pm 66$$-10 + 4\lambda - 144 + 18\lambda + 66 = \pm 66$$22\lambda - 88 = \pm 66$Case $1$: $22\lambda - 88 = 66 \Rightarrow 22\lambda = 154 \Rightarrow \lambda = 7$Case $2$: $22\lambda - 88 = -66 \Rightarrow 22\lambda = 22 \Rightarrow \lambda = 1$Thus,$\lambda = 1, 7$.

For what value of $\lambda$ is the volume of the tetrahedron with vertices having position vectors $\hat{i} - 6\hat{j} + 10\hat{k}$,$-\hat{i} - 3\hat{j} + 7\hat{k}$,$5\hat{i} - \hat{j} + \lambda\hat{k}$,and $7\hat{i} - 4\hat{j} + 7\hat{k}$ equal to $11$ cubic units?

Similar Questions

Statement-$1$: If the points $(1, 2, 2), (2, 1, 2), (2, 2, z)$ and $(1, 1, 1)$ are coplanar,then $z = 2$.
Statement-$2$: If $4$ points $P, Q, R$ and $S$ are coplanar,then the volume of the tetrahedron $PQRS$ is $0$.

If the vectors $2\hat{i}-\hat{j}+3\hat{k}$,$\hat{i}+4\hat{j}+\hat{k}$,and $4\hat{i}+p\hat{j}+\hat{k}$ are coplanar,then $p=$

The sum of the distinct real values of $\mu$,for which the vectors $\mu \hat{i} + \hat{j} + \hat{k}$,$\hat{i} + \mu \hat{j} + \hat{k}$,and $\hat{i} + \hat{j} + \mu \hat{k}$ are coplanar,is

If the vectors $a \hat{i}+\hat{j}+\hat{k}, \hat{i}+b \hat{j}+\hat{k}, \hat{i}+\hat{j}+c \hat{k}$ $(a \neq 1, b \neq 1, c \neq 1)$ are coplanar,then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is

If the vectors $2i - 3j + 4k$, $i + 2j - k$ and $xi - j + 2k$ are coplanar, then $x = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module