Chapter 3, Problem 1CQ

(a)

To determine

Whether a vector can have nonzero magnitude if a component is zero.

Answer to Problem 1CQ

Yes, a vector can have a non-zero magnitude even if a component is zero.

Explanation of Solution

Vector is defined as a quantity having a magnitude and direction. It can be exemplified graphically in two or three dimensions.

Write the expression for the vector $P$ in a $xy$-plane in terms of the component along the axis.

    $\overrightarrow{P}=P_x\widehat{i}+P_y\widehat{j}$

Here, $P_x$ is the component of the vector $\overrightarrow{P}$ along the x-axis and $P_y$ is the component of the vector $\overrightarrow{P}$ along the y-axis.

Consider that the vector is moving along the x-axis. Therefore, if the vector is moving along the x-axis then, its y-component is zero. Hence, the magnitude of the vector is non zero even a component of the vector is zero.

A ball projected vertically upwards is an example in which a vector can have a non-zero magnitude even if a component is zero.

(b)

To determine

Whether a vector can have zero magnitude and a non-zero component.

Answer to Problem 1CQ

No, a vector cannot have zero magnitude and a non–zero component.

Explanation of Solution

Write the expression for the magnitude of the vector.

    $\left|\overrightarrow{P}\right|=\sqrt{{\left(P_x\right)}^2+{\left(P_y\right)}^2}$

Here, $P_x$ is the component of the vector $P$ along the x-axis and $P_y$ is the component of the vector $P$ along the y-axis.

The above equation shows that if any one value of the component is non-zero then the magnitude will also be non-zero.

Therefore, a vector cannot have zero magnitude and a non–zero component.