If the volume of the parallelepiped with over | vedclass

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

Question

(A) Given,the volume of the parallelepiped with coterminous edges $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ is given by the scalar

Vedclass · Accepted Answer

$80$

Vedclass · Answer

(A) Given,the volume of the parallelepiped with coterminous edges $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ is given by the scalar triple product $[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}] = 40 	ext{ cubic units}$.The volume of the parallelepiped with coterminous edges $\overrightarrow{b}+\overrightarrow{c}, \overrightarrow{c}+\overrightarrow{a}, \overrightarrow{a}+\overrightarrow{b}$ is given by the scalar triple product $[\overrightarrow{b}+\overrightarrow{c}, \overrightarrow{c}+\overrightarrow{a}, \overrightarrow{a}+\overrightarrow{b}]$.Using the property of the scalar triple product:$[\overrightarrow{b}+\overrightarrow{c}, \overrightarrow{c}+\overrightarrow{a}, \overrightarrow{a}+\overrightarrow{b}] = (\overrightarrow{b}+\overrightarrow{c}) \cdot ((\overrightarrow{c}+\overrightarrow{a}) 	imes (\overrightarrow{a}+\overrightarrow{b}))$$= (\overrightarrow{b}+\overrightarrow{c}) \cdot (\overrightarrow{c} 	imes \overrightarrow{a} + \overrightarrow{c} 	imes \overrightarrow{b} + \overrightarrow{a} 	imes \overrightarrow{a} + \overrightarrow{a} 	imes \overrightarrow{b})$Since $\overrightarrow{a} 	imes \overrightarrow{a} = 0$,this simplifies to:$= (\overrightarrow{b}+\overrightarrow{c}) \cdot (\overrightarrow{c} 	imes \overrightarrow{a} + \overrightarrow{c} 	imes \overrightarrow{b} + \overrightarrow{a} 	imes \overrightarrow{b})$$= \overrightarrow{b} \cdot (\overrightarrow{c} 	imes \overrightarrow{a}) + \overrightarrow{b} \cdot (\overrightarrow{c} 	imes \overrightarrow{b}) + \overrightarrow{b} \cdot (\overrightarrow{a} 	imes \overrightarrow{b}) + \overrightarrow{c} \cdot (\overrightarrow{c} 	imes \overrightarrow{a}) + \overrightarrow{c} \cdot (\overrightarrow{c} 	imes \overrightarrow{b}) + \overrightarrow{c} \cdot (\overrightarrow{a} 	imes \overrightarrow{b})$$= [\overrightarrow{b} \overrightarrow{c} \overrightarrow{a}] + 0 + 0 + 0 + 0 + [\overrightarrow{c} \overrightarrow{a} \overrightarrow{b}]$$= [\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}] + [\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}] = 2[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$$= 2 	imes 40 = 80 	ext{ cubic units}$.

Similar Questions

$[(a \times b) \times (b \times c), (b \times c) \times (c \times a), (c \times a) \times (a \times b)] = \,$

The value of $\frac{[(\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}), (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}), (\vec{c} \times \vec{a}) \times (\vec{a} \times \vec{b})]}{[\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}]}$ is equal to

If the vector $\overline{c}$ lies in the plane of $\overline{a}$ and $\overline{b}$,where $\overline{a}=\hat{i}-\hat{j}+2\hat{k}$,$\overline{b}=\hat{i}+\hat{j}+\hat{k}$ and $\overline{c}=x\hat{i}-(2-x)\hat{j}-\hat{k}$,then the value of $x$ is

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

Vedclass Test Series

Exam Paper Generator

Online Exam Module