Chapter 3, Problem 1CQ

a. Can a vector have nonzero magnitude if a component is zero? If no, why not? If yes, give an example.

b. Can a vector have zero magnitude and a nonzero component? If no, why not? If yes, give an example.

To determine

A vector have a nonzero magnitude if a component is zero.

Answer to Problem 1CQ

Vector can have nonzero magnitude if a component is zero is explained with an example.

Explanation of Solution

Let, Vector be $\overrightarrow{v}$ and components of $\overrightarrow{v}$ be $v_1,v_2,v_3,\mathrm{.....}{v}_n.$

Write the expression to find the magnitude of vector.

$\left|\overrightarrow{v}\right|=\sqrt{v_1^2+v_2^2+v_3^2+\mathrm{...}+v_n^2}$

So, Magnitude of vector $\left|\overrightarrow{v}\right|$ is zero if and only iff all the components are zero. If any one of the component result with non zero, then vector will have nonzero magnitude.

Example:

Consider the two dimensional vector as follows.

$\overrightarrow{v}=5\text{i + }0\text{j}$

In this vector the y component is 0 but still the magnitude is 5. A vector only has zero magnitude when all its components are 0.

Thus, vector can have nonzero magnitude if a component is zero.

Conclusion:

Hence, vector can have nonzero magnitude if a component is zero is explained with an example.

To determine

A vector have a zero magnitude and a nonzero component.

Answer to Problem 1CQ

Vector cannot have zero magnitude with non zero component is explained.

Explanation of Solution

As explained in part (a), the magnitude of vector is zero if and only iff all the components is zero. If any of the component is nonzero then magnitude of the vector result with nonzero.

Therefore vector cannot have zero magnitude with non zero component.

Example:

Consider a vector as follows.

$\overrightarrow{v}=v_1\text{i}+v_2\text{j}+v_3\text{k}$

Find the magnitude of vector $\overrightarrow{v}$.

$\left|\overrightarrow{v}\right|=\sqrt{v_1^2+v_2^2+v_3^2}$

From the above expression, if $v_1^{}+v_2^{}+v_3^{}$ is non zero value, then magnitude of $\overrightarrow{v}$ will also be non zero.

Conclusion:

Hence, cannot have zero magnitude with non zero component is explained.