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

 Page 4 / 6
${R}_{x}={A}_{x}+\left(–{B}_{x}\right)$

and

${R}_{y}={A}_{y}+\left(–{B}_{y}\right)$

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.

## Phet explorations: vector addition

Learn how to add vectors. Drag vectors onto a graph, change their length and angle, and sum them together. The magnitude, angle, and components of each vector can be displayed in several formats.

## 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

$\begin{array}{lll}{A}_{x}& =& A\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta \\ {B}_{x}& =& B\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta \end{array}$

and

$\begin{array}{lll}{A}_{y}& =& A\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta \\ {B}_{y}& =& B\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta \text{.}\end{array}$

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 $\mathbf{\text{R}}$ :

$\theta ={\text{tan}}^{-1}\left({R}_{y}/{R}_{x}\right).$

## Conceptual questions

Suppose you add two vectors $\mathbf{A}$ and $\mathbf{B}$ . What relative direction between them produces the resultant with the greatest magnitude? What is the maximum magnitude? What relative direction between them produces the resultant with the smallest magnitude? What is the minimum magnitude?

Give an example of a nonzero vector that has a component of zero.

Explain why a vector cannot have a component greater than its own magnitude.

If the vectors $\mathbf{A}$ and $\mathbf{B}$ are perpendicular, what is the component of $\mathbf{A}$ along the direction of $\mathbf{B}$ ? What is the component of $\mathbf{B}$ along the direction of $\mathbf{A}$ ?

## 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 following for path D 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.

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

#### Questions & Answers

what is angular velocity
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an 8.0 capacitor is connected by to the terminals of 60Hz whoes rms voltage is 150v. a.find the capacity reactance and rms to the circuit
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is the study of matter in relation to energy
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Obrian Reply
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aron Reply
power P = Work done per second W/ t. It means the more power, the stronger machine
Sphere
e.g. heart Uses 2 W per beat.
Rohit
A spherica, concave shaving mirror has a radius of curvature of 32 cm .what is the magnification of a persons face. when it is 12cm to the left of the vertex of the mirror
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Shii
1.75cm
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the magnification k = f/(f-d) with focus f = R/2 =16 cm; d =12 cm k = 16/4 =4
Sphere
what do we call velocity
Kings
A weather vane is some sort of directional arrow parallel to the ground that may rotate freely in a horizontal plane. A typical weather vane has a large cross-sectional area perpendicular to the direction the arrow is pointing, like a “One Way” street sign. The purpose of the weather vane is to indicate the direction of the wind. As wind blows pa
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Anurag
What was the previous far point of a patient who had laser correction that reduced the power of her eye by 7.00 D, producing a normal distant vision power of 50.0 D for her?
Jaydie Reply
What is the far point of a person whose eyes have a relaxed power of 50.5 D?
Jaydie
What is the far point of a person whose eyes have a relaxed power of 50.5 D?
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A young woman with normal distant vision has a 10.0% ability to accommodate (that is, increase) the power of her eyes. What is the closest object she can see clearly?
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Ju
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Ju
And don't forget to check out YouTube videos on the subject. Videos offer a different visual way to learn easier.
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