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

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.

By signing up, you agree to our Terms of Service and Privacy Policy

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.

By signing up, you agree to our Terms of Service and Privacy Policy

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

By signing up, you agree to our Terms of Service and Privacy Policy

- Objects A and B are at the origin. If object A moves to #(-6 ,-5 )# and object B moves to #(-1 ,12 )# over #3 s#, what is the relative velocity of object B from the perspective of object A? Assume that all units are denominated in meters.
- A projectile is shot from the ground at an angle of #( pi)/3 # and a speed of #9 /4 m/s#. Factoring in both horizontal and vertical movement, what will the projectile's distance from the starting point be when it reaches its maximum height?
- Objects A and B are at the origin. If object A moves to #(-3 ,5 )# and object B moves to #(-6 ,8 )# over #3 s#, what is the relative velocity of object B from the perspective of object A? Assume that all units are denominated in meters.
- What is the dot product of # (2i -3j + 4k)# and #(- 4 i - 5 j + 2)#?
- A projectile is shot at an angle of #pi/12 # and a velocity of #3 6 m/s#. How far away will the projectile land?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7