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.

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°

abc

= 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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APPEARS IN

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