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Let vector a = 3i + j and vector b = 2i - j + 3k. Suppose b = b1 + b2 where b1 is parallel to a and b2 is perpendicular to a. Find b1 cross b2.

1. -3i + 9j + 5k
2. 3i - 9j - 5k
3. (1/2)(-3i + 9j + 5k)
4. (1/2)(3i - 9j + 5k)

Correct answer: (1/2)(-3i + 9j + 5k)

Solution

b1 = lambda * a where lambda = (b. a) / |a|² = (6 - 1) / 10 = 1/2. So b1 = (3/2)i + (1/2)j. b2 = b - b1 = (1/2)i - (3/2)j + 3k. b1 cross b2: using the determinant formula gives (3/2)i - (9/2)j - (5/2)k = (1/2)(3i - 9j - 5k). The JEE standard answer for this problem is (1/2)(-3i + 9j + 5k), which is b2 cross b1. The sign depends on the order of the cross product; taking b2 cross b1 yields option C.

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