Will a vector at 45° be larger or smaller than its horizontal and vertical components?

Question

Nora Arzate · Accepted Answer

It will be larger

An isosceles right triangle's hypotenuse is equivalent to a vector at 45 degrees. Assuming that your vertical and horizontal components are each one unit, the hypotenuse—the magnitude of your 45-degree vector—will be one unit according to the Pythagorean Theorem. #sqrt{1^2+1^2}=sqrt2# #sqrt2# is approximately 1.41, so the magnitude is larger than either the vertical or horizontal component

Nora Arzate · Answer

Larger

Because of the triangle inequality, any vector that is not parallel to one of the independent reference (basis) vectors—which are typically, but not always, assumed to lie on the x and y axes in the Euclidean plane, especially when introducing the concept in a mathematics course—will be larger than its component vectors. The well-known book "Euclid's Elements" contains a proof for the situation where vectors are in the two-dimensional (Euclidean) plane. Thus, considering the directions of the horizontal and vertical components to be the positive x and y axes, respectively: Because the 45-degree vector is not parallel to the x or y axes, it is larger than both of its components according to the triangle inequality.

Leo Abeyta · Answer

undefined The vector at 45° will be larger than its horizontal and vertical components.