Name: Cross Product
Uploaded: 2019-07-19
Duration: 4 min 16 s
Description: The cross product a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a particular direction and magnitude. Watch this vi...

Question

Find the volume of the parallelepiped whose adjacent edges are represented by

2 \vec{a}, - \vec{b} and 3 \vec{c}

, where

\vec{a} = \hat{i} - \hat{j} + 2 \hat{k}, \vec{b} = 3 \hat{i} + 4 \hat{j} - 5 \hat{k}, and \vec{c} = 2 \hat{i} - \hat{j} + 3 \hat{k}

.

Accepted Answer

Given vectors: $\vec{a} = \hat{i} - \hat{j} + 2 \hat{k}, \vec{b} = 3 \hat{i} + 4 \hat{j} - 5 \hat{k}, and \vec{c} = 2 \hat{i} - \hat{j} + 3 \hat{k}$

Now, $2 \vec{a} = 2 (\hat{i} - \hat{j} + 2 \hat{k}) = 2 \hat{i} - 2 \hat{j} + 4 \hat{k}$

$- \vec{b} = - 3 \hat{i} - 4 \hat{j} + 5 \hat{k}$

$3 \vec{c} = 3 (2 \hat{i} - \hat{j} + 3 \hat{k}) = 6 \hat{i} - 3 \hat{j} + 9 \hat{k}$

The volume of the parallelepiped whose adjacent edges are represented by $2 \vec{a}, - \vec{b} and 3 \vec{c}$ is $|2 \vec{a} \cdot (- \vec{b} \times 3 \vec{c})|$

Now, $- \vec{b} \times 3 \vec{c} = |\begin{array}{r} \hat{i} & \hat{j} & \hat{k} \\ - 3 & - 4 & 5 \\ 6 & - 3 & 9 \end{array}| = (- 36 + 15) \hat{i} - (- 27 - 30) \hat{j} + (9 + 24) \hat{k} = - 21 \hat{i} + 57 \hat{j} + 33 \hat{k}$

And also, $2 \vec{a} \cdot (- \vec{b} \times 3 \vec{c}) = (2 \hat{i} - 2 \hat{j} + 4 \hat{k}) \cdot (- 21 \hat{i} + 57 \hat{j} + 33 \hat{k})$

$= 2 (- 21) - 2 (57) + 4 (33)$

$= - 42 - 114 + 132$

$= - 24$

Therefore, the volume of parallelepiped is $|- 24|$ or $24$ cubic units.

Find the volume of the parallelepiped whose adjacent edges are represented by $2 \vec{a}, - \vec{b} and 3 \vec{c}$ , where $\vec{a} = \hat{i} - \hat{j} + 2 \hat{k}, \vec{b} = 3 \hat{i} + 4 \hat{j} - 5 \hat{k}, and \vec{c} = 2 \hat{i} - \hat{j} + 3 \hat{k}$ .

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

Find the volume of the parallelepiped whose adjacent edges are represented by 2a→,-b→ and 3c→, where a→=i^-j^+2k^,b→=3i^+4j^-5k^, and c→=2i^-j^+3k^.

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

Find the volume of the parallelepiped whose adjacent edges are represented by $2 \vec{a}, - \vec{b} and 3 \vec{c}$ , where $\vec{a} = \hat{i} - \hat{j} + 2 \hat{k}, \vec{b} = 3 \hat{i} + 4 \hat{j} - 5 \hat{k}, and \vec{c} = 2 \hat{i} - \hat{j} + 3 \hat{k}$ .