If the volume of a tetrahedron having bar{i}+ | vedclass

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

Question

(A) The volume of a tetrahedron with coterminous edges $\vec{a}, \vec{b}, \vec{c}$ is given by $V = \frac{1}{6} |[\vec{a} \vec{b} \vec{c}]|$.Given $\v

Vedclass · Accepted Answer

$x^2+4 x-12=0$

Vedclass · Answer

(A) The volume of a tetrahedron with coterminous edges $\vec{a}, \vec{b}, \vec{c}$ is given by $V = \frac{1}{6} |[\vec{a} \vec{b} \vec{c}]|$.Given $\vec{a} = \bar{i}+2 \bar{j}-3 \bar{k}$, $\vec{b} = 2 \bar{i}+\bar{j}-3 \bar{k}$, and $\vec{c} = 3 \bar{i}-\bar{j}+p \bar{k}$.The scalar triple product is $[\vec{a} \vec{b} \vec{c}] = \begin{vmatrix} 1 & 2 & -3 \ 2 & 1 & -3 \ 3 & -1 & p \end{vmatrix}$.Expanding the determinant: $1(p-3) - 2(2p+9) - 3(-2-3) = p-3 - 4p-18 + 15 = -3p-6$.Given volume $V = 2$, so $\frac{1}{6} |-3p-6| = 2$.$|-3p-6| = 12$.This implies $-3p-6 = 12$ or $-3p-6 = -12$.Case $1$: $-3p = 18 \implies p = -6$.Case $2$: $-3p = -6 \implies p = 2$.The values of $p$ are $2$ and $-6$.The quadratic equation whose roots are $2$ and $-6$ is $(x-2)(x+6) = 0$.$x^2+6x-2x-12 = 0 \implies x^2+4x-12 = 0$.

If the volume of a tetrahedron having $\bar{i}+2 \bar{j}-3 \bar{k}$, $2 \bar{i}+\bar{j}-3 \bar{k}$, and $3 \bar{i}-\bar{j}+p \bar{k}$ as its coterminous edges is $2$, then the values of $p$ are the roots of the equation

Similar Questions

The value of $[\vec{a}-\vec{b} \quad \vec{b}-\vec{c} \quad \vec{c}-\vec{a}]$ is equal to

$i \cdot (j \times k) + j \cdot (k \times i) + k \cdot (i \times j) = $

Let $p, q, r$ be three non-coplanar vectors and $b = p \times q$. If $a, b, c$ denote the coterminous edges of a parallelepiped, then its height with the base having $a$ and $c$ is

Let $\vec{a} = \hat{i} + 2\hat{j} + 4\hat{k}$,$\vec{b} = \hat{i} + \lambda\hat{j} + 4\hat{k}$,and $\vec{c} = 2\hat{i} + 4\hat{j} + (\lambda^2 - 1)\hat{k}$ be coplanar vectors. Then the non-zero vector $\vec{a} \times \vec{c}$ is:

Let $a, b$ and $c$ be distinct non-negative numbers. If the vectors $a\hat{i} + a\hat{j} + c\hat{k}$,$\hat{i} + \hat{k}$ and $c\hat{i} + c\hat{j} + b\hat{k}$ are coplanar,then $c = \dots$

Vedclass Test Series

Exam Paper Generator

Online Exam Module