Determine whether each statement is true or false. Provide a brief justification.
a. Two lines with parallel direction vectors willnot intersect.
b. Two linesthat are perpendicularin R3 willhave perpendicular direction vectors.
c. Two linesthat havenon-paralleldirection vectors willbe skewlines.
a. Statement (a) is true. Two lines with parallel direction vectors will not intersect.
To determine if two lines are parallel, we can compare their direction vectors. If the direction vectors of two lines are scalar multiples of each other, the lines are parallel. If the lines are parallel, then they will never intersect, and vice versa. This is a result of the fact that parallel lines have the same slope and will never converge or intersect.
b. Statement (b) is false. Two lines that are perpendicular in R3 do not necessarily have perpendicular direction vectors.
In three-dimensional space (R3), two lines can be perpendicular to each other even if their direction vectors are not perpendicular. The direction vectors of two lines determine their slopes but not their perpendicularity. Two lines can have different slopes but still intersect at a right angle.
c. Statement (c) is true. Two lines that have non-parallel direction vectors will be skew lines.
Skew lines are lines in three-dimensional space that do not intersect and are not parallel. If two lines have different direction vectors, they will not be parallel and hence they cannot intersect. Therefore, they are skew lines.