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Prove that the Volume of a Parallelopiped with Coterminal Edges as a,b,c Hence Find the Volume of the Parallelopiped with Coterminal Edges i+j, j+k - Mathematics and Statistics

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Prove that the volume of a parallelopiped with coterminal edges as  ` bara ,bar b , barc `

Hence find the volume of the parallelopiped with coterminal edges  `bar i+barj, barj+bark `

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Solution

Let, `bar a, bar b and bar c` be the position vectors of points A, B and C respectively with respect to origin O.
Complete the parallelopiped as shown in the figure with ` bar(OA) , bar(OB) and bar(OC)` as its coterminus edges.
AP is a perpendicular drawn to the plane of `bar b and bar c` . Let, θ be the angle made by AP with OA.
Volume of parallelopiped = (Area of parallelogram OCDB) x (height)
Now, area of parallelogram OCDB = `|barb xxbarc| ….(i)`

Height of parallelepiped =l(AP)

`=l(OA) cos theta`

`= |bar(OA)| cos theta`

`= |bara| cos theta .....(ii)`

From (i) and (ii) we get,

volume of parallelepiped =`|bara||barbxxbarc|costheta`

`= bar a.(barb xx barc)`

volume of parallelepiped=`[bara barb barc]`

`Let bar a = hati + hatj ,bar b = hatj + hatk , c = hati + hatk`

`[bar a bar b barc]=|[1,1,0],[0,1,1],[1,0,1]|`

=1(1-0)-1(0-1)+0

=1+1

=2

Concept: Scalar Triple Product of Vectors
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