Question 12

$$\text{Let }\vec a=3\hat i-\hat j+2\hat k,\quad\vec b=\vec a\times(\hat i-2\hat k)\text{ and } \vec c=\vec b\times\hat k.\text{Then the projection of } (\vec c-2\hat j)\text{ on } \vec a \text{ is:}$$

a=3i-j+2k, b=a×(i-2k), c=b×k. Find projection of (c-2j) on a.

b = (3i-j+2k)×(i-2k) = |i j k; 3 -1 2; 1 0 -2| = i(2-0)-j(-6-2)+k(0+1) = 2i+8j+k

c = b×k = (2i+8j+k)×k = |i j k; 2 8 1; 0 0 1| = i(8)-j(2)+k(0) = 8i-2j

c-2j = 8i-4j. Projection on a: (8i-4j)·(3i-j+2k)/|a| = (24+4+0)/√14 = 28/√14 = 2√14.

The correct answer is Option 1: 2√14.

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