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

Let $a, b, c$ be distinct non-negative numbers. If the vectors $a\hat{i} + a\hat{j} + c\hat{k}$,$\hat{i} + \hat{k}$,and $c\hat{i} + c\hat{j} + b\hat{k}$ lie in a plane,then $c$ is

If $\overline{a}, \overline{b}$ and $\overline{c}$ are unit coplanar vectors,then the scalar triple product $[2 \overline{a}-\overline{b}, 2 \overline{b}-\overline{c}, 2 \overline{c}-\overline{a}]$ has the value

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number,then for what value of $\lambda$ does the equation $[\lambda(\vec{a} + \vec{b}), \lambda^2\vec{b}, \lambda\vec{c}] = [\vec{a}, \vec{b} + \vec{c}, \vec{b}]$ hold?

Let $\vec{a} = \hat{i} - \hat{j}$,$\vec{b} = \hat{j} - \hat{k}$,and $\vec{c} = \hat{k} - \hat{i}$. If $\vec{d}$ is a unit vector such that $\vec{a} \cdot \vec{d} = 0 = [\vec{b} \, \vec{c} \, \vec{d}]$,then find $\vec{d}$.

What will be the volume of the parallelepiped whose coterminous edges are given by the vectors $a = i - j + k$,$b = i - 3j + 4k$,and $c = 2i - 5j + 3k$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module