Vectors: True or False Questions

Question

Determine which of the following statements are true and which are false.

(a) There exist vectors v, w ∈ R³ such that ||v|| = 3, ||w|| = 3, and ||v + w|| = 7.

(b) There exist vectors v, w ∈ R² such that ||v|| = 2, ||w|| = 3, and v · w = −1.

(c) If v, w ∈ Rⁿ are unit vectors, then |v · w| ≤ 1.

(d) If v, w ∈ R² are vectors with v · w > 0, then the angle between v and w is acute (i.e., less than 90 degrees)

Answer 1

(a) False. By the triangular inequality, ||v + w|| ≤ ||v|| + ||w||, which is not fulfilled for the given values.

(b) True. v · w = ||v||.||w||.cos(a) where a is the angle between v and w. between Replacing the values we get v · w = 6 cos(a). The only way we get v · w = −1 is by choosing cos(a) = (-1/6), which is doable by tuning a = acos(-1/6).

(c) True. By Cauchy-Schwarz, |v · w|^2 ≤ |v · v|.| w · w|. But |v · v| = | w · w| = 1 since both vectors are unit ones. Hence replacing in the inequality we get |v · w|^2 ≤ (1)(1) = 1 and square rooting both sides we get |v · w| ≤ 1

(d) True. v · w = ||v||.||w||.cos(a) > 0 where a is the angle between v and w, means that cos(a) has to be positive since norms are positive. Since cos(a) > 0 means that -90 < a < 90.