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Find the volume of the parallelepiped with one vertex at (1, -1, -1), and adjacent vertices at...

Question:

Find the volume of the parallelepiped with one vertex at {eq}(1, -1, -1) {/eq}, and adjacent vertices at {eq}(-4, -7, 6) {/eq}, {eq}(8, -7, 1) {/eq}, and {eq}(4, -2, 4) {/eq}.

The Volume of a Parallelopiped:

Let the vectors {eq}\vec{a}\;=\;a_{1}\hat{i}\;+\;a_{2}\hat{j}\;+\;a_{3}\hat{k} {/eq}

{eq}\vec{b}\;=\;b_{1}\hat{i}\;+\;b_{2}\hat{j}\;+\;b_{3}\hat{k} {/eq} and

{eq}\vec{c}\;=\;c_{1}\hat{i}\;+\;c_{2}\hat{j}\;+\;c_{3}\hat{k} {/eq}

are the adjacent sides of the parallelopiped then its volume is given by

{eq}[\;\vec{a} \;\vec{b} \;\vec{c}\;]\;=\;\begin{vmatrix} a_{1} &a_{2} &a_{3} \\ b_{1} &b_{2} &b_{3} \\ c_{1} &c_{2} &c_{3} \end{vmatrix} {/eq}

Answer and Explanation:

Given {eq}O (1, -1, -1) , P (-4,-7,6), Q (8,-7,1) {/eq} and {eq}R (4,-2,4) {/eq}

Then sides are,

{eq}\vec{OP}\;=\; {/eq} Coordinate of {eq}P {/eq} - Coordinate of {eq}O\;=\;(-4\;-\;1)\hat{i}\;+\;(-7\;+\;1)\hat{j}\;+\;(6\;+\;1)\hat{k} {/eq}

{eq}\vec{OP}\;=\;-5\hat{i}\;-\;6\vec{j}\;+\;7\vec{k} {/eq}

{eq}\vec{OQ}\;=\; {/eq} Coordinate of {eq}Q {/eq} - Coordinate of {eq}O\;=\;(8\;-\;1)\hat{i}\;+\;(-7\;+\;1)\hat{j}\;+\;(1\;+\;1)\hat{k} {/eq}

{eq}\vec{OQ}\;=\;7\vec{i}\;-\;6\vec{j}\;+\;2\vec{k} {/eq}

{eq}\vec{OR}\;=\; {/eq} Coordinate of {eq}R {/eq} - Coordinate of {eq}O\;=\;(4\;-\;1)\hat{i}\;+\;(-2\;+\;1)\hat{j}\;+\;(4\;+\;1)\hat{k} {/eq}

{eq}\vec{OR}\;=\;3\vec{i}\;-\;\vec{j}\;+\;5\vec{k} {/eq}

Therefore

Volume of parallelopiped {eq}=\; \begin{vmatrix} -5 &-6 &7\\ 7 &-6 &2 \\ 3 &-1 &5 \end{vmatrix} \;= \;237 {/eq}

Volume = 237 cubic units.


Learn more about this topic:

Vectors in Two & Three Dimensions
Vectors in Two & Three Dimensions

from NDA Exam Preparation & Study Guide

Chapter 13 / Lesson 4
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