2.3 Vector addition and subtraction: analytical methods  (Page 4/6)

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