Question

If `veca + vecb, vecb + vecc` and `vecc + veca` are coterminous edges of a parallelepiped then its volume is ______.

Options

• `3[(veca, vecc, vecb)]`

• 0

• `2[(veca, vecb, vecc)]`

• `4[(vecb, veca, vecc)]`

Answer 1

If `veca + vecb, vecb + vecc` and `vecc + veca` are coterminous edges of a parallelepiped then its volume is `underlinebb(2[(veca, vecb, vecc)])`.

Explanation:

∵ `veca + vecb, vecb + vecc` and `vecc + veca` are coterminous edges of a parallelepiped.

Then, its volume (v) = `[(veca + vecb, vecb + vecc, vecc + veca)]`

We know, scalar triple product

`[(veca, vecb, vecc)] = veca.(vecb xx vecc) ≡ (veca xx vecb).vecc`

Consider `[(veca + vecb, vecb + vecc, vecc + veca)]`

= `(veca + vecb).{(vecb + vecc) xx (vecc + veca)}`

= `(veca + vecb).{(vecb xx vecc) + (vecb xx veca) + (vecc xx vecc) + (vecc xx veca)}`

= `(veca + vecb).{(vecb xx vecc) + (vecb xx veca) + (vecc xx veca)}`  ...`(∵ vecc xx vecc = 0)`

= `veca.(vecb xx vecc) + veca.(vecb xx veca) + veca.(vecc xx veca) + vecb.(vecb xx vecc) + vecb.(vecb xx veca) + vecb.(vecc xx veca)`

= `[(veca, vecb, vecc)] + [(veca, vecb, veca)] + [(veca, vecc, veca)] + [(vecb, vecb, vecc)] + [(vecb, vecb, veca)] + [(vecb, vecc, veca)]`

= `[(veca, vecb, vecc)] + [(vecb, vecc, veca)]`

= `2[(veca, vecb, vecc)]`