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

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$8[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$

(C) The volume of a parallelepiped with coterminous edges $\vec{u}, \vec{v}, \vec{w}$ is given by the scalar triple product $[\vec{u} \vec{v} \vec{w}] = (\vec{u} \times \vec{v}) \cdot \vec{w}$.Given edges are $2\vec{a}, 2\vec{b}, 2\vec{c}$.Volume $= [2\vec{a} \ 2\vec{b} \ 2\vec{c}]$$= (2\vec{a} \times 2\vec{b}) \cdot 2\vec{c}$$= 4(\vec{a} \times \vec{b}) \cdot 2\vec{c}$$= 8(\vec{a} \times \vec{b}) \cdot \vec{c}$$= 8[\vec{a} \vec{b} \vec{c}]$.

Class 12 Mathematics Vector Algebra Scalar triple product and their applications

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The volume of a parallelepiped whose coterminous edges are $2 \overrightarrow{a}, 2 \overrightarrow{b}, 2 \overrightarrow{c}$ is:

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A
$2[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$
B
$4[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$
C
$8[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$
D
$[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$

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If the three coterminous edges of a parallelepiped are represented by the vectors $(a - b)$,$(b - c)$,and $(c - a)$,find its volume.

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Statement-$1$: If the points $(1, 2, 2), (2, 1, 2), (2, 2, z)$ and $(1, 1, 1)$ are coplanar,then $z = 2$.
Statement-$2$: If $4$ points $P, Q, R$ and $S$ are coplanar,then the volume of the tetrahedron $PQRS$ is $0$.

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If $\overline{a}$ and $\overline{c}$ are unit vectors inclined at $\frac{\pi}{3}$ with each other and $(\overline{a} \times (\overline{b} \times \overline{c})) \cdot (\overline{a} \times \overline{c}) = 5$,then the value of $5[\overline{a} \overline{b} \overline{c}] = $

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If $a, b, c$ are non-negative distinct numbers and $a \hat{\imath}+a \hat{\jmath}+c \hat{k}$,$\hat{\imath}+\hat{k}$ and $c \hat{\imath}+c \hat{\jmath}+b \hat{k}$ are coplanar vectors,then

EasyMHT CET 2020

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Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{j}-\hat{k}.$ If $\overrightarrow{c}$ is a vector such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3$,then $\vec{a} \cdot(\vec{b} \times \vec{c})$ is equal to :

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The volume of a parallelepiped whose coterminous edges are $2 \overrightarrow{a}, 2 \overrightarrow{b}, 2 \overrightarrow{c}$ is:

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If the three coterminous edges of a parallelepiped are represented by the vectors $(a - b)$,$(b - c)$,and $(c - a)$,find its volume.

Statement-$1$: If the points $(1, 2, 2), (2, 1, 2), (2, 2, z)$ and $(1, 1, 1)$ are coplanar,then $z = 2$.Statement-$2$: If $4$ points $P, Q, R$ and $S$ are coplanar,then the volume of the tetrahedron $PQRS$ is $0$.

If $\overline{a}$ and $\overline{c}$ are unit vectors inclined at $\frac{\pi}{3}$ with each other and $(\overline{a} \times (\overline{b} \times \overline{c})) \cdot (\overline{a} \times \overline{c}) = 5$,then the value of $5[\overline{a} \overline{b} \overline{c}] = $

If $a, b, c$ are non-negative distinct numbers and $a \hat{\imath}+a \hat{\jmath}+c \hat{k}$,$\hat{\imath}+\hat{k}$ and $c \hat{\imath}+c \hat{\jmath}+b \hat{k}$ are coplanar vectors,then

Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{j}-\hat{k}.$ If $\overrightarrow{c}$ is a vector such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3$,then $\vec{a} \cdot(\vec{b} \times \vec{c})$ is equal to :

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Statement-$1$: If the points $(1, 2, 2), (2, 1, 2), (2, 2, z)$ and $(1, 1, 1)$ are coplanar,then $z = 2$.
Statement-$2$: If $4$ points $P, Q, R$ and $S$ are coplanar,then the volume of the tetrahedron $PQRS$ is $0$.