Let the vectors vec{a}, vec{b}, vec{c} repres | vedclass

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

Question

(C) The volume of a parallelepiped with coterminous edges $\vec{a}, \vec{b}, \vec{c}$ is given by the scalar triple product $[\vec{a}, \vec{b}, \vec{c

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(C) The volume of a parallelepiped with coterminous edges $\vec{a}, \vec{b}, \vec{c}$ is given by the scalar triple product $[\vec{a}, \vec{b}, \vec{c}] = V$. The volume of the new parallelepiped with edges $\vec{a}, \vec{b}+\vec{c}, \vec{a}+2\vec{b}+3\vec{c}$ is given by the scalar triple product $[\vec{a}, \vec{b}+\vec{c}, \vec{a}+2\vec{b}+3\vec{c}]$. Using the properties of the scalar triple product,we can express this as the determinant of the coefficients of the vectors $\vec{a}, \vec{b}, \vec{c}$: $[\vec{a}, \vec{b}+\vec{c}, \vec{a}+2\vec{b}+3\vec{c}] = \begin{vmatrix} 1 & 0 & 0 \ 0 & 1 & 1 \ 1 & 2 & 3 \end{vmatrix} [\vec{a}, \vec{b}, \vec{c}]$. Calculating the determinant: $1(1 	imes 3 - 1 	imes 2) - 0 + 0 = 1(3 - 2) = 1$. Thus,the volume is $1 	imes V = V$.

Let the vectors $\vec{a}, \vec{b}, \vec{c}$ represent three coterminous edges of a parallelepiped of volume $V$. Then the volume of the parallelepiped,whose coterminous edges are represented by $\vec{a}, \vec{b}+\vec{c}$ and $\vec{a}+2\vec{b}+3\vec{c}$ is equal to $..........\,V$.

Similar Questions

If $\vec{a}=\hat{i}-\hat{k}, \vec{b}=x \hat{i}+\hat{j}+(1-x) \hat{k}$ and $\vec{c}=y \hat{i}+x \hat{j}+(1+x-y) \hat{k}$,then $[\vec{a} \vec{b} \vec{c}]$ depends on

The volume of a tetrahedron whose vertices are $A \equiv (-1, 2, 3)$,$B \equiv (3, -2, 1)$,$C \equiv (2, 1, 3)$,and $D \equiv (-1, -2, 4)$ is

The volume of the parallelepiped whose coterminous edges are $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$, $\vec{b} = 3\hat{i} - \hat{j} - \hat{k}$, and $\vec{c} = \hat{j} + 3\hat{k}$ is:

The number of distinct real values of $\lambda$,for which the vectors $-\lambda^2 \hat{i}+\hat{j}+\hat{k}$,$\hat{i}-\lambda^2 \hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-\lambda^2 \hat{k}$ are coplanar,is

$i \cdot(j \times k)+j \cdot(k \times i)+k \cdot(j \times i)$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module