M427L Quiz 2 Solution
Question. Working in the plane R2, consider the line Lgiven by y=mx + 2, with ma constant.
Determine
(i) a unit vector parallel to L;
(ii) a unit vector orthogonal to L.
Solution
(i) Note that (0,2) and (1, m + 2) are two points on the line L. Being two points on L, the
vector from one point to the other is parallel to L. So i+mj(the vector from (0,2) to
(1, m + 2)) is parallel to L. The unit vector in the direction of i+mjis
1
√m2+ 1i+m
√m2+ 1j.
(ii) Note that −mi+jsatisfies that
(−mi+j)·(i+mj) = 0.
Both vectors in the left-hand side are non-zero, so they are orthonogal. Since i+mjis
parallel to L,−mi+jis therefore orthogonal to L. The unit vector in the direction of
−mi+jis
−m
√m2+ 1i+1
√m2+ 1j.
1