7.5 Show that a. (b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors, a, b and c.

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Vishal Baghel | Contributor-Level 10

4 months ago

Let $\overset{?}{c}$ = $\overset{?}{b}$ , $\overset{?}{a}$ = $\overset{?}{a}$ , $\overset{?}{O A}$ = $\overset{?}{b}$

Let $\overset{?}{O B}$ be a unit vector perpendicular to both b and c. Hence $\overset{?}{c}$ and a have the same direction

Now $\overset{?}{O C}$ = bc $\overset{?}{n}$ $\overset{?}{n}$

= bc $\overset{?}{b} \times \overset{?}{c}$ $\sin ? θ$ = bc $\overset{?}{n}$

Now $\sin ? 90 °$ )= a. (bc $\overset{?}{n}$ ) = abccos $\overset{?}{n}$ ? = abccos0° = abc = Volume of the parallelepiped

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