Chapter 3, Problem 8Q

Can the magnitude of a vector ever (a) equal, or (b) be less than, one of its components?

Solution

To Determine:

Whether the magnitude of a vector can be equal or less than one of its components.

Solution:

The magnitude may be equal but not less than one of its components.

Explanation:

For a vector’s magnitude to be equal to one of its components, that component should be equal to the magnitude and the others equal to zero. On the other hand, the magnitude of a vector will never be less than one of its components.

We need to know how to calculate the magnitude of a vector. If we define $\overrightarrow{V}$ as:

  $\overrightarrow{V}=x\widehat{i}+y\widehat{j}+z\widehat{k}$

Its magnitude is:

Let suppose:

  $y=0,z=0$

Then:

  $\left|\overrightarrow{V}\right|=\sqrt{x^2+0+0}=x$

Even if the value of the component is negative, for finding the resultant, it will be squared and hence, the resultant would be greater than 0 the value of resultant cannot be reduced below the value of the component.

Conclusions:

The vector magnitude is always greater or equal to one of its components but never less than one of them.