Question

True or false (give a reason if true or a counterexample if false):

(a) If u is perpendicular (in three dimensions) to v and w, those vectors v and w are parallel. "

(b) If u is perpendicular to v and w, then u is perpendicular to v + 2 w,

(c) If u and v are perpendicular unit vectors then II u - v" = ,.,fi,

Answer 1

solution:

A) False. The vectors could be skew. Example: u = <0,0,1>, v = <1,0,0>, and w = <0,1,0>

B) False. This is usually true, but there are examples when the vectors could cancel each other out Example: u = <0,0,1>, v = <2,0,0>, and w = <-1,0,0>

C) True. A unit vector must have a magnitude of 1, and if two vectors are perpendicular to each other, then you know the angle between them is 90 degrees. Whenever you add or subtract vectors, you can think about the resulting vector as a line that connects the two vectors to complete a triangle, which in this case, would be the hypotenuse. Since the triangle has side lengths of 1, the hypotenuse (||v-w|| or ||v+w||) will equal sqrt(2)