What are the direction ratios of the line?
Direction: Consider the following for the two (02) items that follow :<br>Let L:x+y+z+4=0=2x-y-z+8 be a line and P:x+2y+3z+1=0 be a plane.
- A. \langle 2, 1, -1 \rangle
- B. \langle 0, -1, 2 \rangle
- C. \langle 0, 1, -1 \rangle ✓
- D. \langle 2, 3, -3 \rangle
Correct Answer: C. \langle 0, 1, -1 \rangle
Explanation
The line L is formed by the intersection of the planes x+y+z+4=0 and 2x-y-z+8=0. The direction vector of L is the cross product of their normals \vec{n_1} = \langle 1, 1, 1 \rangle and \vec{n_2} = \langle 2, -1, -1 \rangle. Thus, \vec{n_1} \times \vec{n_2} = \langle 1(-1) - 1(-1), 1(2) - 1(-1), 1(-1) - 1(2) \rangle = \langle 0, 3, -3 \rangle. This is proportional to \langle 0, 1, -1 \rangle.
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