Chapter 4, Problem 1RQ

Review Question 4.1 When does a vector have a positive scalar component? When does a vector have a negative scalar component?

To determine

The condition under which a vector has a positive scalar component and a negative scalar component.

Answer to Problem 1RQ

Solution:

When the vector component is in the positive direction of the axis, the scalar component is positive.

When the vector component is in the negative direction of the axis, the scalar component is negative.

Explanation of Solution

Introduction:

The scalar component of vector is determined by the x-axis and the y-axis, which are perpendicular to each other. There is no need to use the vector component of $\overrightarrow{F}$. It can be replaced with perpendicular forces $F_x$ and $F_y$.

Explanation:

Consider the first case, when a vector is in the positive x-direction,

The following steps are used to draw the diagram,

(1) Draw an arrow that represents the vector $\overrightarrow{F}$.

(2) Now, draw the x-axis and the y-axis, which are perpendicular to each other.

(3) Resolve the vector into components.

Here, $F$ is the magnitude of the force vector, $\theta$ is the angle with respect to the positive x-axis, $F_x$ is the force component in the positive x-direction, and $F_y$ is the force component in the positive y-direction.

Resolve the magnitude of force $F$ as,

$F_x=F\cos \theta$

And,

$F_y=F\sin \theta$

Therefore, when the vector component is in the positive direction of the axis, the scalar component is positive.

Consider the second case, when a vector is in the negative x-direction,

Here, $F$ is the magnitude of the force vector, $\theta$ is the angle with respect to the negative x-axis, $F_x$ is the force component in the negative x-direction, and $F_y$ is the force component in the positive y-direction.

Find the values of the scalar component as,

$F_x=-F\cos \theta$

And,

$F_y=F\sin \theta$

Suppose $F_y$ is the force component in the negative y-direction, then,

$F_y=-F\sin \theta$

Here, the negative sign shows the direction of the vector.

Therefore, when the vector component is in the negative direction of the axis, the scalar component is negative.

Conclusion:

Thus, if the vector component is in the positive direction of the axis, then the scalar component is positive and, if the vector component is in the negative direction of the axis, the scalar component is negative.