3.4.1 Vector Math

You learned in Section 3.2 how to add vectors graphically, but it can be a tedious problem in geometry and trigonometry to find precise values for the magnitude and direction of the resultant. The addition and subtraction of vectors become much easier if we use components and unit vectors.

To see this, let’s evaluate the vector sum $\vec{D} = \vec{A} + \vec{B} + \vec{C}$ . To begin, write this sum in terms of the components of each vector:

\begin{array}{l} \vec{D} = D_{x} î + D_{y} \hat{j} = \vec{A} + \vec{B} + \vec{C} \\ = (A_{x} î + A_{y} \hat{j}) + (B_{x} î + B_{y} \hat{j}) + (C_{x} î + C_{y} \hat{j}) \end{array}

(3.8)

We can group together all the x-components and all the y-components on the right side, in which case Equation 3.8 is

(D_{x}) î + (D_{y}) ĵ = (A_{x} + B_{x} + C_{x}) î + (A_{y} + B_{y} + C_{y}) ĵ

(3.9)

Comparing the x- and y-components on the left and right sides of Equation 3.9, we find:

\begin{array}{l} D_{x} = A_{x} + B_{x} + C_{x} \\ D_{y} = A_{y} + B_{y} + C_{y} \end{array}

(3.10)

Stated in words, Equation 3.10 says that we can perform vector addition by adding the x-components of the individual vectors to give the x-component of the resultant and by adding the y-components of the individual vectors to give the y-component of the resultant. This method of vector addition is called algebraic addition.

EXAMPLE 3.6 Using algebraic addition to find a displacement

Example 3.1 was about a bird that flew 100 m to the east, then 50 m to the northwest. Use the algebraic addition of vectors to find the bird’s net displacement.

VISUALIZE FIGURE 3.19 shows displacement vectors $\vec{A} = (100 m, east)$ and $\vec{B} = (50 m, northwest)$ . We draw vectors tip-to-tail to add them graphically, but it’s usually easier to draw them all from the origin if we are going to use algebraic addition.

The figure illustrates the net displacement labeled vector C resulting from the addition of two displacement vectors labeled vector "A" and vector B on the x y coordinate plane.

FIGURE 3.19 Full Alternative Text

SOLVE To add the vectors algebraically we must know their components. From the figure these are seen to be

\begin{array}{l} \vec{A} = 100 î m \\ \vec{B} = (- 50 cos 45 ° î + 50 sin 45 ° \hat{j}) m = (- 35.3 î + 35.3 \hat{j}) m \end{array}

Notice that vector quantities must include units. Also notice, as you would expect from the figure, that $\vec{C}$ has a negative x-component. Adding $\vec{A}$ and $\vec{B}$ by components gives

\begin{matrix} \vec{C} & = & \vec{A} + \vec{B} = 100 î m + (- 35.3 î + 35.3 ĵ) m \\ = & (100 m - 35.3 m) î + (35.3 m) ĵ = (64.7 î + 35.3 ĵ) m \end{matrix}

This would be a perfectly acceptable answer for many purposes. However, we need to calculate the magnitude and direction of $\vec{C}$ if we want to compare this result to our earlier answer. The magnitude of $\vec{C}$ is

C = \sqrt{C_{x}^{2} + v_{y}^{2}} = \sqrt{(64.7 {m)}^{2} + (35.3 {m)}^{2}} = 74 m

The angle $ϕ$ , as defined in FIGURE 3.19, is

ϕ = {tan}^{- 1} (\frac{C_{y}}{C_{x}}) = {tan}^{- 1} (\frac{35.3 m}{64 .7 m}) = 29 °

Thus $\vec{C} = (74 m, 29 ° north of east)$ , in perfect agreement with Example 3.1.

Vector subtraction and the multiplication of a vector by a scalar, using components, are very much like vector addition. To find $\vec{R} = \vec{P} - \vec{Q}$ we would compute

\begin{array}{l} R_{x} = P_{x} - Q_{x} \\ R_{y} = P_{y} - Q_{y} \end{array}

(3.11)

Similarly, $\vec{T} = c \vec{S}$ would be

\begin{array}{l} T_{x} = c S_{x} \\ T_{y} = c S_{y} \end{array}

(3.12)

In other words, a vector equation is interpreted as meaning: Equate the x-components on both sides of the equals sign, then equate the y-components, and then the z-components. Vector notation allows us to write these three equations in a compact form.