Question

For the next pairs of lines, determine if they are parallel, perpendicular or neither of those.line through (0,1,0) with direction (5,5,4), and line through (0,1,0)with direction (1,−2,1).line through (312,524,345) with direction (1,1,1), and line through(123,323,553) with direction (3,3,3).line through points (3,6,5) and (5,4,2), and line passing through(0,0,0) with direction (30,−30,−45).

Answer 1

Given

a) line with direction (5, 5, 4); line with direction (1, -2, 1)

b) line with direction (1, 1, 1); line with direction (3, 3, 3)

c) line through points (3, 6, 5) and (5, 4, 2); line with direction (30, -30, -45)

Find

whether the lines are parallel, perpendicular, or neither

Solution

a) The directions are different, so the lines are not parallel. The dot product of the direction vectors is 5·1 + 5·(-2) + 4·1 = 5-10+4 = -1 ≠ 0, so the lines are not perpendicular. They are neither.

b) When a factor of 3 is removed from the second direction vector, it is identical to the first: (3, 3, 3)/3 = (1, 1, 1). These lines are parallel.

c) The direction vector of the first line can be found by subtracting the first point from the second: (5, 4, 2) - (3, 6, 5) = (2, -2, -3). Removing a factor of 15 from the direction vector of the second line gives (30, -30, -45)/15 = (2, -2, -3). This direction vector is the same as that for the first line. These lines are parallel.