If the volume of the parallelepiped whose cot | vedclass

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

Question

(A) Given that the volume of the parallelepiped formed by vectors $\bar{a}, \bar{b}, \bar{c}$ is $12$. Thus,the scalar triple product is $[\bar{a} \ba

Vedclass · Accepted Answer

$4 	ext{ (units)}^3$

Vedclass · Answer

(A) Given that the volume of the parallelepiped formed by vectors $\bar{a}, \bar{b}, \bar{c}$ is $12$. Thus,the scalar triple product is $[\bar{a} \bar{b} \bar{c}] = 12$. The volume of a tetrahedron with coterminous edges $\bar{u}, \bar{v}, \bar{w}$ is given by $\frac{1}{6} |[\bar{u} \bar{v} \bar{w}]|$. Here,the edges are $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$. We calculate the scalar triple product: $[\bar{a}+\bar{b} \quad \bar{b}+\bar{c} \quad \bar{c}+\bar{a}] = (\bar{a}+\bar{b}) \cdot ((\bar{b}+\bar{c}) 	imes (\bar{c}+\bar{a}))$ $= (\bar{a}+\bar{b}) \cdot (\bar{b} 	imes \bar{c} + \bar{b} 	imes \bar{a} + \bar{c} 	imes \bar{c} + \bar{c} 	imes \bar{a})$ Since $\bar{c} 	imes \bar{c} = 0$,this simplifies to: $= (\bar{a}+\bar{b}) \cdot (\bar{b} 	imes \bar{c} + \bar{b} 	imes \bar{a} + \bar{c} 	imes \bar{a})$ $= \bar{a} \cdot (\bar{b} 	imes \bar{c}) + \bar{a} \cdot (\bar{b} 	imes \bar{a}) + \bar{a} \cdot (\bar{c} 	imes \bar{a}) + \bar{b} \cdot (\bar{b} 	imes \bar{c}) + \bar{b} \cdot (\bar{b} 	imes \bar{a}) + \bar{b} \cdot (\bar{c} 	imes \bar{a})$ $= [\bar{a} \bar{b} \bar{c}] + 0 + 0 + 0 + 0 + [\bar{b} \bar{c} \bar{a}]$ $= [\bar{a} \bar{b} \bar{c}] + [\bar{a} \bar{b} \bar{c}] = 2[\bar{a} \bar{b} \bar{c}] = 2(12) = 24$. Therefore,the volume of the tetrahedron is $\frac{1}{6} 	imes 24 = 4 	ext{ (units)}^3$.

If the volume of the parallelepiped whose coterminous edges are along the vectors $\bar{a}, \bar{b}, \bar{c}$ is $12$,then the volume of the tetrahedron whose coterminous edges are $\bar{a}+\bar{b}, \bar{b}+\bar{c}$ and $\bar{c}+\bar{a}$ is

Similar Questions

If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i}$, and $\vec{c} = c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k}$ with $c_1 = 1$ and $c_2 = 2$, then find the value of $c_3$ such that $\vec{a}$, $\vec{b}$, and $\vec{c}$ are coplanar.

Given the vectors $\vec x = 3i - 6j - k$,$\vec y = i + 4j - 3k$,and $\vec z = 3i - 4j - 12k$,find the projection of the vector $\vec x \times \vec y$ onto the vector $\vec z$.

If the volume of a parallelepiped with coterminous edges $4 \hat{i} + 5 \hat{j} + \hat{k}$, $-\hat{j} + \hat{k}$, and $3 \hat{i} + 9 \hat{j} + p \hat{k}$ is $34$ cubic units, then $p$ is equal to:

Consider the vectors $u = a \hat{i} + b \hat{j} + c \hat{k}$,$v = a^2 \hat{i} + b^2 \hat{j} + c^2 \hat{k}$ and $w = a^3 \hat{i} + b^3 \hat{j} + c^3 \hat{k}$. These vectors are coplanar if and only if

Vedclass Test Series

Exam Paper Generator

Online Exam Module