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Question
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 `
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
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