Question

The magnitude of a vector can never be less than the magnitude of one of its components.

True or False
Explain your answer with detail​

Answer 1

True

Step-by-step explanation:

We can actually prove this to be the case.

Let’s say we have a vector V with components X, Y, and Z. Let’s assume that X has a greater magnitude than V.

|V| = (X^2 + Y^2 + Z^2)^(1/2) < X

x^2 + Y^2 + Z^2 < X^2

Y^2 + Z^2 < 0

But Y^2 and Z^2 are strictly >=0, so there is no way they can be less than 0.

This proof trivially extends to any dimension vector.

One way to internalize this is to consider a right angle triangle.

The vector is the line from A to B and the components are AC and CB. There is no way AB can ever be shorter than AC or CB, (otherwise you could never reach it!)