3.3.3 Components

You learned in Chapters 1 and 2 to give the kinematic variable $v_{x}$ a positive sign if the velocity vector $\vec{v}$ points toward the positive end of the x-axis, a negative sign if $\vec{v}$ points in the negative x-direction. We need to extend this idea to vectors in general.

Suppose vector $\vec{A}$ has been decomposed into component vectors $\vec{A}_{x}$ and $\vec{A}_{y}$ parallel to the coordinate axes. We can describe each component vector with a single number called the component. The x-component and y-component of vector $\vec{A}$ , denoted $A_{x}$ and $A_{y}$ , are determined as follows:

TACTICS BOX 3.1 Determining the components of a vector

1 The absolute value $| A_{x} |$ of the x-component $A_{x}$ is the magnitude of the component vector $\vec{A}_{x}$ .

2 The sign of $\vec{A}_{x}$ is positive if $\vec{A}_{x}$ points in the positive x-direction (right), negative if $\vec{A}_{x}$ points in the negative x-direction (left).

3 The y-component $A_{y}$ is determined similarly.

Workbook Exercises 10–18

In other words, the component $A_{x}$ tells us two things: how big $\vec{A}_{x}$ is and, with its sign, which end of the axis $\vec{A}_{x}$ points toward. FIGURE 3.11 shows three examples of determining the components of a vector.

The figure illustrates components of two displacement vectors labeled vector "A" and vector B using two separate x y coordinate planes.

FIGURE 3.11 Full Alternative Text

Full Alternative Text

NOTE Beware of the somewhat confusing terminology. $\vec{A}_{x}$ and $\vec{A}_{y}$ are called component vectors, whereas $A_{x}$ and $A_{y}$ are simply called components. The components $A_{x}$ and $A_{y}$ are just numbers (with units), so make sure you do not put arrow symbols over the components.

We will frequently need to decompose a vector into its components. We will also need to “reassemble” a vector from its components. In other words, we need to move back and forth between the geometric and the component representations of a vector. FIGURE 3.12 shows how this is done.

The figure illustrates components of two velocity vectors labeled vector "A" and vector B using two separate x y coordinate planes.

FIGURE 3.12 Full Alternative Text

Each decomposition requires that you pay close attention to the direction in which the vector points and the angles that are defined.

If a component vector points left (or down), you must manually insert a minus sign in front of the component, as was done for $B_{y}$ in FIGURE 3.12.
The role of sines and cosines can be reversed, depending upon which angle is used to define the direction. Compare $A_{x}$ and $B_{x}$ .
The angle used to define direction is almost always between $0 °$ and $90 °$ , so you must take the inverse tangent of a positive number. Use absolute values of the components, as was done to find angle $ϕ$ (Greek phi) in FIGURE 3.12.

EXAMPLE 3.3 Finding the components of an acceleration vector

Seen from above, a hummingbird’s acceleration is ( $6.0 m / s^{2}, 30 °$ south of west). Find the x- and y-components of the acceleration vector $\vec{a}$ .

VISUALIZE It’s important to draw vectors. FIGURE 3.13 establishes a map-like coordinate system with the x-axis pointing east and the y-axis north. Vector $\vec{a}$ is then decomposed into components parallel to the axes. Notice that the axes are “acceleration axes” with units of acceleration, not xy-axes, because we’re measuring an acceleration vector.

The figure illustrates the components of an acceleration vector labeled vector “a” in the third quadrant of the x y coordinate plane.

FIGURE 3.13 Full Alternative Text

SOLVE The acceleration vector points to the left (negative x-direction) and down (negative y-direction), so the components $a_{x}$ and $a_{y}$ are both negative:

\begin{array}{l} a_{x} = - a cos 30 ° = - (6.0 m / s^{2}) cos 30 ° = - 5.2 m / s^{2} \\ a_{y} = - a sin 30 ° = - (6.0 m / s^{2}) sin 30 ° = - 3.0 m / s^{2} \end{array}

REVIEW The units of $a_{x}$ and $a_{y}$ are the same as the units of vector $\vec{a}$ . Notice that we had to insert the minus signs manually by observing that the vector points left and down.

EXAMPLE 3.4 Finding the direction of motion

FIGURE 3.14 shows a car’s velocity vector $\vec{v}$ . Determine the car’s speed and direction of motion.

The figure illustrates a velocity vector labeled v in the second quadrant of the x y coordinate plane.

FIGURE 3.14 Full Alternative Text

VISUALIZE FIGURE 3.15 shows the components $v_{x}$ and $v_{y}$ and defines an angle $θ$ with which we can specify the direction of motion.

The figure illustrates the components of a velocity vector labeled vector V in the second quadrant of a coordinate plane.

FIGURE 3.15 Full Alternative Text

SOLVE We can read the components of $\vec{v}$ directly from the axes: $v_{x} = - 6.0 m / s$ and $v_{y} = 4.0 m / s$ . Notice that $v_{x}$ is negative. This is enough information to find the car’s speed $v$ , which is the magnitude of $\vec{v} :$

v = \sqrt{v_{x}^{2} + v_{y}^{2}} = \sqrt{{(- 6.0 m/s)}^{2} + {(4.0 m/s)}^{2}} = 7.2 m / s

From trigonometry, angle $θ$ is

θ = {tan}^{- 1} (\frac{v_{y}}{| v_{x} |}) = {tan}^{- 1} (\frac{4.0 m/s}{6.0 m/s}) = 34 °

The absolute value signs are necessary because $v_{x}$ is a negative number. The velocity vector $\vec{v}$ can be written in terms of the speed and the direction of motion as

\vec{v} = (7.2 m / s, 34 ° a b o v e t h e n e g a t i v e x - axis)

3.3.3 Components

TACTICS BOX 3.1 Determining the components of a vector

FIGURE 3.11 Determining the components of a vector.

FIGURE 3.12 Moving between the geometric representation and the component representation.

EXAMPLE 3.3 Finding the components of an acceleration vector

FIGURE 3.13 Decomposition of $\vec{a}$ .

EXAMPLE 3.4 Finding the direction of motion

FIGURE 3.14 The velocity vector $\vec{v}$ of Example 3.4.

FIGURE 3.15 Decomposition of $\vec{v}$ .