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AIEEE 2002

JEE Main 2002 Mathematics question on Vector Algebra

If V⃗=2i^+j^−k^\vec{V} = 2\hat{i} + \hat{j} - \hat{k} and W⃗=i^+3k^\vec{W} = \hat{i} + 3\hat{k}. If U⃗\vec{U} is a unit vector, then the maximum value of the scalar triple product [U⃗  V⃗  W⃗][\vec{U} \; \vec{V} \; \vec{W}] is
  1. A.
    −1-1
  2. B.
    10+6\sqrt{10} + \sqrt{6}
  3. C.
    59\sqrt{59}
  4. D.
    60\sqrt{60}
Show answer and solution

Correct option: C

Solution

The scalar triple product [U V W] = U · (V × W). First compute V × W:

V × W = |i j k; 2 1 -1; 1 0 3| = i(1·3 - (-1)·0) - j(2·3 - (-1)·1) + k(2·0 - 1·1)
= i(3) - j(7) + k(-1) = 3i - 7j - k

The maximum value of U · (V × W) over all unit vectors U is simply |V × W| (achieved when U is parallel to V × W).

|V × W| = √(9 + 49 + 1) = √59

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