The volume of a parallelepiped whose cotermin | vedclass

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

Question

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

Vedclass · Accepted Answer

$\frac{1}{\sqrt{2}}$

Vedclass · Answer

(A) The volume of a parallelepiped with coterminous edges $\hat{a}, \hat{b}, \hat{c}$ is given by the scalar triple product $|[\hat{a} \hat{b} \hat{c}]|$.The square of the volume is given by the determinant of the Gram matrix:$V^2 = \begin{vmatrix} \hat{a} \cdot \hat{a} & \hat{a} \cdot \hat{b} & \hat{a} \cdot \hat{c} \ \hat{b} \cdot \hat{a} & \hat{b} \cdot \hat{b} & \hat{b} \cdot \hat{c} \ \hat{c} \cdot \hat{a} & \hat{c} \cdot \hat{b} & \hat{c} \cdot \hat{c} \end{vmatrix}$Since $\hat{a}, \hat{b}, \hat{c}$ are unit vectors,$\hat{a} \cdot \hat{a} = \hat{b} \cdot \hat{b} = \hat{c} \cdot \hat{c} = 1$. Given $\hat{a} \cdot \hat{b} = \hat{b} \cdot \hat{c} = \hat{c} \cdot \hat{a} = \frac{1}{2}$,we have:$V^2 = \begin{vmatrix} 1 & 1/2 & 1/2 \ 1/2 & 1 & 1/2 \ 1/2 & 1/2 & 1 \end{vmatrix}$Calculating the determinant:$V^2 = 1(1 - 1/4) - 1/2(1/2 - 1/4) + 1/2(1/4 - 1/2)$$V^2 = 1(3/4) - 1/2(1/4) + 1/2(-1/4)$$V^2 = 3/4 - 1/8 - 1/8 = 3/4 - 2/8 = 3/4 - 1/4 = 2/4 = 1/2$Therefore,the volume $V = \sqrt{1/2} = \frac{1}{\sqrt{2}}$.

The volume of a parallelepiped whose coterminous edges are represented by unit vectors $\hat{a}, \hat{b}, \hat{c}$ such that $\hat{a} \cdot \hat{b} = \hat{b} \cdot \hat{c} = \hat{c} \cdot \hat{a} = \frac{1}{2}$ is:

Similar Questions

Let $S$ be the set of all $(\lambda, \mu)$ for which the vectors $\lambda \hat{i} - \hat{j} + \hat{k}$,$\hat{i} + 2\hat{j} + \mu \hat{k}$ and $3\hat{i} - 4\hat{j} + 5\hat{k}$,where $\lambda - \mu = 5$,are coplanar,then $\sum_{(\lambda, \mu) \in S} 80(\lambda^2 + \mu^2)$ is equal to :

Let $V = 2i + j - k$ and $W = i + 3k$. If $U$ is a unit vector,then the maximum value of the scalar triple product $[U V W]$ is

If the position vectors of the vertices $A, B$ and $C$ are $6i$,$6j$ and $k$ respectively with respect to the origin $O$,then the volume of the tetrahedron $OABC$ is

Let $\overrightarrow{A} = \hat{i} + \hat{j} + \hat{k}$,$\overrightarrow{B} = \hat{i}$,and $\overrightarrow{C} = C_1\hat{i} + C_2\hat{j} + C_3\hat{k}$. If $C_2 = -1$ and $C_3 = 1$,then to make the three vectors coplanar:

Let $a, b$ and $c$ be three vectors. Then the scalar triple product $[a, b, c]$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module