Lines in space practice problems
Parametric and symmetric equations of lines, and whether two lines are parallel, intersecting or skew. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Find parametric and symmetric equations of the line through \( \displaystyle P(6, 5, 2) \) and \( \displaystyle Q(-4, -4, -1) \).
Find parametric and symmetric equations of the line through \( \displaystyle P(5, -3, -3) \) and \( \displaystyle Q(-4, -2, 0) \).
Find parametric and symmetric equations of the line through \( \displaystyle P(2, 4, -6) \) and \( \displaystyle Q(1, -1, -4) \).
Find parametric and symmetric equations of the line through \( \displaystyle P(0, -5, -5) \) and \( \displaystyle Q(-1, -2, -4) \).
Find parametric and symmetric equations of the line through \( \displaystyle P(0, 4, 0) \) and \( \displaystyle Q(-2, -1, -1) \).
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle - t - 1, - t, - t - 2 \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle - 3 s - 3, 1 - s, 2 s - 4 \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
Find parametric and symmetric equations of the line through \( \displaystyle P(2, 6, -3) \) and \( \displaystyle Q(6, 0, 4) \).
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle 2 - 3 t, 4 - 3 t, 3 t - 3 \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle 5, 8 - s, 2 s - 8 \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle 1 - 3 t, 2 t - 1, 3 - 2 t \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle - s - 3, - s, 1 \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle t - 1, 3, 2 t \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle 2 s, 3, 3 s + 1 \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.