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AIEEE 2008

JEE Main 2008 Mathematics question on Vector Algebra

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vector a^\hat{a}, b^\hat{b}, c^\hat{c} such that a^⋅b^=b^⋅c^=c^⋅a^=12\hat{a} \cdot \hat{b} = \hat{b} \cdot \hat{c} = \hat{c} \cdot \hat{a} = \frac{1}{2}. Then, the volume of the parallelopiped is
  1. A.
    12\frac{1}{\sqrt{2}} cu unit
  2. B.
    122\frac{1}{2\sqrt{2}} cu unit
  3. C.
    32\frac{\sqrt{3}}{2} cu unit
  4. D.
    13\frac{1}{\sqrt{3}} cu unit
Show answer and solution

Correct option: A

Solution

The volume is |[â b̂ ĉ]|, i.e., the scalar triple product. Compute its square using the Gram determinant:

[â b̂ ĉ]² = det of the matrix of dot products:
|1 1/2 1/2|
|1/2 1 1/2|
|1/2 1/2 1 |

Expanding: 1(1 - 1/4) - (1/2)(1/2 - 1/4) + (1/2)(1/4 - 1/2)
= 1(3/4) - (1/2)(1/4) + (1/2)(-1/4)
= 3/4 - 1/8 - 1/8
= 3/4 - 1/4 = 1/2

So Volume = √(1/2) = 1/√2.

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