0.3 2.4 vector addition: analytical methods  (Page 3/4)

Step 3. To get the magnitude $R$ of the resultant, use the Pythagorean theorem:

Step 4. To get the direction of the resultant:

The following example illustrates this technique for adding vectors using perpendicular components.

Adding vectors using analytical methods

Add the vector $\mathbf{A}$ to the vector $\mathbf{B}$ shown in [link] , using perpendicular components along the x - and y -axes. The x - and y -axes are along the east–west and north–south directions, respectively. Vector $\mathbf{A}$ represents the first leg of a walk in which a person walks $\text{53}\text{.}\text{0 m}$ in a direction $\text{20}\text{.}0\text{º}$ north of east. Vector $\mathbf{B}$ represents the second leg, a displacement of $\text{34}\text{.}\text{0 m}$ in a direction $\text{63}\text{.}0\text{º}$ north of east.

Strategy

The components of $\mathbf{A}$ and $\mathbf{B}$ along the x - and y -axes represent walking due east and due north to get to the same ending point. Once found, they are combined to produce the resultant.

Solution

Following the method outlined above, we first find the components of $\mathbf{A}$ and $\mathbf{B}$ along the x - and y -axes. Note that $A=\mathrm{53.0\; m}$ , ${\theta }_{\text{A}}=\mathrm{20.0º}$ , $B=\mathrm{34.0\; m}$ , and ${\theta }_{\text{B}}=\mathrm{63.0º}$ . We find the x -components by using $A_x=A\text{cos}\theta$ , which gives

and

Similarly, the y -components are found using $A_y=A\text{sin}{\theta }_{\mathrm{A}}$ :

and

The x - and y -components of the resultant are thus

and

Now we can find the magnitude of the resultant by using the Pythagorean theorem:

so that

Finally, we find the direction of the resultant:

Thus,

Discussion

This example illustrates the addition of vectors using perpendicular components. Vector subtraction using perpendicular components is very similar—it is just the addition of a negative vector.

Subtraction of vectors is accomplished by the addition of a negative vector. That is, $\mathbf{A}-\mathbf{B}\equiv \mathbf{A}+(\mathbf{–B})$ . Thus, the method for the subtraction of vectors using perpendicular components is identical to that for addition . The components of $\mathbf{\text{–B}}$ are the negatives of the components of $\mathbf{B}$ . The x - and y -components of the resultant $\mathbf{A}-\text{B = R}$ are thus

and

and the rest of the method outlined above is identical to that for addition. (See [link] .)

Analyzing vectors using perpendicular components is very useful in many areas of physics, because perpendicular quantities are often independent of one another. The next module, Projectile Motion , is one of many in which using perpendicular components helps make the picture clear and simplifies the physics.

Summary

• The analytical method of vector addition and subtraction involves using the Pythagorean theorem and trigonometric identities to determine the magnitude and direction of a resultant vector.
• The steps to add vectors $\mathbf{A}$ and $\mathbf{B}$ using the analytical method are as follows:

  Step 1: Determine the coordinate system for the vectors. Then, determine the horizontal and vertical components of each vector using the equations

  $\begin{array}{lll}{A}_x& =& A\text{cos}\theta \\ B_x& =& B\text{cos}\theta \end{array}$

  and

  $\begin{array}{lll}{A}_y& =& A\text{sin}\theta \\ B_y& =& B\text{sin}\theta \text{.}\end{array}$

  Step 2: Add the horizontal and vertical components of each vector to determine the components $R_x$ and $R_y$ of the resultant vector, $\mathbf{\text{R}}$ :

  $R_x=A_x+B_x$

  and

  $R_y=A_y+B_{y.}$

  Step 3: Use the Pythagorean theorem to determine the magnitude, $R$ , of the resultant vector $\mathbf{\text{R}}$ :

  $R=\sqrt{R_x^2+R_y^2}.$

  Step 4: Use a trigonometric identity to determine the direction, $\theta$ , of $\mathbf{\text{R}}$ :

  $\theta ={\text{tan}}^{-1}(R_y/R_x).$

Conceptual questions

Problems&Exercises

Find the following for path C in [link] : (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.

(a) 1.56 km

(b) 120 m east

Find the north and east components of the displacement from San Francisco to Sacramento shown in [link] .

North-component 87.0 km, east-component 87.0 km

Solve the following problem using analytical techniques: Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements $\mathbf{A}$ and $\mathbf{B}$ , as in [link] , then this problem asks you to find their sum $\mathbf{R}=\mathbf{A}+\mathbf{B}$ .)

Note that you can also solve this graphically. Discuss why the analytical technique for solving this problem is potentially more accurate than the graphical technique.

A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as $\mathbf{A},$ $\mathbf{B},$ and $\mathbf{C}$ in [link] , and then correctly calculates the length and orientation of the fourth side $\mathbf{D}$ . What is his result?

Suppose a pilot flies $\text{40}\text{.}\text{0 km}$ in a direction $\text{60º}$ north of east and then flies $\text{30}\text{.}\text{0 km}$ in a direction $\text{15º}$ north of east as shown in [link] . Find her total distance $R$ from the starting point and the direction $\theta$ of the straight-line path to the final position. Discuss qualitatively how this flight would be altered by a wind from the north and how the effect of the wind would depend on both wind speed and the speed of the plane relative to the air mass.