If the volume of a tetrahedron whose contermi | vedclass

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

Question

(C) The volume of a tetrahedron with edges $\overline{u}, \overline{v}, \overline{w}$ is given by $V = \frac{1}{6} |[\overline{u} \overline{v} \overli

Vedclass · Accepted Answer

$72$ cubic units

Vedclass · Answer

(C) The volume of a tetrahedron with edges $\overline{u}, \overline{v}, \overline{w}$ is given by $V = \frac{1}{6} |[\overline{u} \overline{v} \overline{w}]|$.Given edges are $\overline{a}+\overline{b}, \overline{b}+\overline{c}, \overline{c}+\overline{a}$.So,$24 = \frac{1}{6} |(\overline{a}+\overline{b}) \cdot ((\overline{b}+\overline{c}) 	imes (\overline{c}+\overline{a}))|$.$144 = |(\overline{a}+\overline{b}) \cdot (\overline{b} 	imes \overline{c} + \overline{b} 	imes \overline{a} + \overline{c} 	imes \overline{c} + \overline{c} 	imes \overline{a})|$.Since $\overline{c} 	imes \overline{c} = 0$,we have $144 = |(\overline{a}+\overline{b}) \cdot (\overline{b} 	imes \overline{c} + \overline{b} 	imes \overline{a} + \overline{c} 	imes \overline{a})|$.Expanding the scalar triple product: $144 = |[\overline{a} \overline{b} \overline{c}] + [\overline{a} \overline{b} \overline{a}] + [\overline{a} \overline{c} \overline{a}] + [\overline{b} \overline{b} \overline{c}] + [\overline{b} \overline{b} \overline{a}] + [\overline{b} \overline{c} \overline{a}]|$.Terms like $[\overline{a} \overline{b} \overline{a}]$ are $0$ because two vectors are identical.Thus,$144 = |[\overline{a} \overline{b} \overline{c}] + [\overline{b} \overline{c} \overline{a}]|$.Since $[\overline{a} \overline{b} \overline{c}] = [\overline{b} \overline{c} \overline{a}]$,we get $144 = 2 |[\overline{a} \overline{b} \overline{c}]|$.Therefore,$|[\overline{a} \overline{b} \overline{c}]| = 72$,which is the volume of the parallelepiped.

If the volume of a tetrahedron whose conterminous edges are $\overline{a}+\overline{b}, \overline{b}+\overline{c}, \overline{c}+\overline{a}$ is $24$ cubic units,then the volume of the parallelepiped whose coterminous edges are $\overline{a}, \overline{b}, \overline{c}$ is

Similar Questions

If $\overrightarrow{a} \cdot \overrightarrow{b} = 1, \overrightarrow{b} \cdot \overrightarrow{c} = 2$ and $\overrightarrow{c} \cdot \overrightarrow{a} = 3$,then the value of $[\vec{a} \times(\vec{b} \times \vec{c}), \vec{b} \times(\vec{c} \times \vec{a}), \vec{c} \times(\vec{b} \times \vec{a})]$ is

If $\overrightarrow x = 3\hat i - 6\hat j - \hat k$,$\overrightarrow y = \hat i + 4\hat j - 3\hat k$ and $\overrightarrow z = 3\hat i - 4\hat j - 12\hat k$,then the magnitude of the projection of $\overrightarrow x \times \overrightarrow y$ on $\overrightarrow z$ is

If $a, b, c$ are non-coplanar vectors and $\lambda$ is a real number,then $[\lambda(a + b), \lambda^2 b, \lambda c] = [a, b + c, b]$ for

For any non-zero vectors $a, b, c$,$a \cdot[(b+c) \times(a+b+c)] = \ldots .$

Let $\vec V = 2\hat i + \hat j - \hat k$,$\vec W = \hat i + 3\hat k$,and $|\vec U| = 2$. If $\vec U$ is a vector in the $x-y$ plane,then the greatest value of $([\vec U \vec V \vec W])^2$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module