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Show That A. (B × C) is Equal in Magnitude to the Volume of the Parallelepiped Formed on the Three Vectors, A, B And C. - Physics

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

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Solution 1

A parallelepiped with origin O and sides ab, and c is shown in the following figure

Volume of the given parallelepiped = abc

`vecOC =  veca`

`vecOB =  vecb`

`vecOC = vecc`

Let `hatn` be a unit vector perpendicular to both b and c. Hence, `hatn` and a have the same direction.

`:. vecb xx vecc =  bc sin theta hatn`

`=bc  sin 90^@ hatn`

`=bc  sin 90^@ hatn`

`= bc  hatn`

`veca.(vecb xx vecc)`

`= a.(bc  hatn)`

abc cosθ `hatn`

abc cos 0°


= Volume of the parallelepiped

Solution 2

Let a parallelopiped be formed on the three vectors.

`vec(OA) = veca`, `vec(OB) = vecb` and `vec(OC) = vecC`

Now `vecb xx vecc = bc  sin 90^@ hatn = bc  hatn`

where `hatn` is unit vector along `vecOA` perpendicular to the plane containing `vecb` and `vecc`

Now `veca.(vecb xx vecc) = veca.bc  hatn`

`=(a)(bc) cos 0^@`

= abc

which is equal in magnitude to the volume of the parallelopiped


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NCERT Class 11 Physics Textbook
Chapter 7 System of Particles and Rotational Motion
Q 5 | Page 178
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