Which of the following are the direction ratios of the line of intersection of the given planes?
For the following two (02) items: A plane P is <strong>PARALLEL</strong> to the line having direction ratios (1, 3, 2) and contains the line of intersection of the planes 6x+4y-5z=2 and x-2y+3z=0.
- A. (2, 23, 16)
- B. \langle 2, -23, -16 \rangle ✓
- C. \langle 2, 3, 2 \rangle
- D. \langle -1, 3, -2 \rangle
Correct Answer: B. \langle 2, -23, -16 \rangle
Explanation
The direction ratios of the line of intersection are given by the cross product of the normal vectors of the two planes. Let \vec{n_1} = \langle 6, 4, -5 \rangle and \vec{n_2} = \langle 1, -2, 3 \rangle. Then \vec{d} = \vec{n_1} \times \vec{n_2} = \hat{i}(12 - 10) - \hat{j}(18 - (-5)) + \hat{k}(-12 - 4) = 2\hat{i} - 23\hat{j} - 16\hat{k}. Thus, the direction ratios are \langle 2, -23, -16 \rangle.
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