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

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R x = A x + ( – B x ) size 12{R rSub { size 8{x} } =A rSub { size 8{x} } +-B rSub { size 8{x} } } {}

and

R y = A y + ( – B y ) size 12{R rSub { size 8{y} } =A rSub { size 8{y} } +-B rSub { size 8{y} } } {}

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.

In this figure, the subtraction of two vectors A and B is shown. A red colored vector A is inclined at an angle theta A to the positive of x axis. From the head of vector A a blue vector negative B is drawn. Vector B is in west of south direction. The resultant of the vector A and vector negative B is shown as a black vector R from the tail of vector A to the head of vector negative B. The resultant R is inclined to x axis at an angle theta below the x axis. The components of the vectors are also shown along the coordinate axes as dotted lines of their respective colors.
The subtraction of the two vectors shown in [link] . The components of –B size 12{B} {} are the negatives of the components of B size 12{B} {} . The method of subtraction is the same as that for addition.

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.

Vector Addition

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 A size 12{A} {} and B size 12{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

    A x = A cos θ B x = B cos θ alignl { stack { size 12{A rSub { size 8{x} } =A"cos"θ} {} #B rSub { size 8{x} } =B"cos"θ {} } } {}

    and

    A y = A sin θ B y = B sin θ . alignl { stack { size 12{A rSub { size 8{y} } =A" sin"θ} {} #B=B suby " sin "θ {} } } {}

    Step 2: Add the horizontal and vertical components of each vector to determine the components R x size 12{R rSub { size 8{x} } } {} and R y size 12{R rSub { size 8{y} } } {} of the resultant vector, R size 12{R} {} :

    R x = A x + B x size 12{R rSub { size 8{x} } =A rSub { size 8{x} } +B rSub { size 8{x} } } {}

    and

    R y = A y + B y . size 12{R rSub { size 8{y} } =A rSub { size 8{y} } +B rSub { size 8{y} } } {}

    Step 3: Use the Pythagorean theorem to determine the magnitude, R size 12{R} {} , of the resultant vector R size 12{R} {} :

    R = R x 2 + R y 2 . size 12{R= sqrt {R rSub { size 8{x} } rSup { size 8{2} } +R rSub { size 8{y} } rSup { size 8{2} } } } {}

    Step 4: Use a trigonometric identity to determine the direction, θ size 12{θ} {} , of R size 12{R} {} :

    θ = tan − 1 ( R y / R x ) . size 12{θ="tan" rSup { size 8{ - 1} } \( R rSub { size 8{y} } /R rSub { size 8{x} } \) } {}

Conceptual questions

Suppose you add two vectors A size 12{A} {} and B size 12{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 A size 12{A} {} and B size 12{B} {} are perpendicular, what is the component of A size 12{A} {} along the direction of B size 12{B} {} ? What is the component of B size 12{B} {} along the direction of A size 12{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 map of city is shown. The houses are in form of square blocks of side one hundred and twenty meter each. Four paths A B C and D are shown in different colors. The path c shown as blue extends to one block towards north, then five blocks towards east and then two blocks towards south then one block towards west and one block towards north and finally three blocks towards west. It is asked to find out the total distance traveled the magnitude and the direction of the displacement from start to finish for path C.
The various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side.

(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] .

A map of northern California with a circle with a radius of one hundred twenty three kilometers centered on San Francisco. Sacramento lies on the circumference of this circle in a direction forty-five degrees north of east from San Francisco.

North-component 87.0 km, east-component 87.0 km
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Source:  OpenStax, Sample chapters: openstax college physics for ap® courses. OpenStax CNX. Oct 23, 2015 Download for free at http://legacy.cnx.org/content/col11896/1.9
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