Chapter 5.1, Problem 8PP

To determine

Whether a vector could be shorter than one of its components.

Whether a vector could be equal in length to one of its components.

Explanation of Solution

Introduction:

A vector $\overrightarrow{v}$ in 2-dimensional space can be described as:

  $\overrightarrow{v}=v_x\widehat{i}+v_y\widehat{j}$

Here, $v_x$ is the horizontal component and $v_y$ is the vertical component.

The magnitude of the vector $\overrightarrow{v}$ can be obtained by:

  $\left|\overrightarrow{v}\right|=\sqrt{v_x{}^2+v_y{}^2}$

Even if one of the components is negative, for calculating the magnitude of the vector, the square of the components is done.

Thus, the vector can be shorter than one of its components.

If other component is zero, then only the vector would be equal in length to its component.