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Learning objectives

By the end of this section, you will be able to:

  • Understand the rules of vector addition and subtraction using analytical methods.
  • Apply analytical methods to determine vertical and horizontal component vectors.
  • Apply analytical methods to determine the magnitude and direction of a resultant vector.

The information presented in this section supports the following AP® learning objectives and science practices:

  • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2)

Analytical methods of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, because vectors are still represented by arrows for easy visualization. However, analytical methods are more concise, accurate, and precise than graphical methods, which are limited by the accuracy with which a drawing can be made. Analytical methods are limited only by the accuracy and precision with which physical quantities are known.

Resolving a vector into perpendicular components

Analytical techniques and right triangles go hand-in-hand in physics because (among other things) motions along perpendicular directions are independent. We very often need to separate a vector into perpendicular components. For example, given a vector like A size 12{A} {} in [link] , we may wish to find which two perpendicular vectors, A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} , add to produce it.

In the given figure a dotted vector A sub x is drawn from the origin along the x axis. From the head of the vector A sub x another vector A sub y is drawn in the upward direction. Their resultant vector A is drawn from the tail of the vector A sub x to the head of the vector A sub y at an angle theta from the x axis. On the graph a vector A, inclined at an angle theta with x axis is shown. Therefore vector A is the sum of the vectors A sub x and A sub y.
The vector A size 12{A} {} , with its tail at the origin of an x , y -coordinate system, is shown together with its x - and y -components, A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} . These vectors form a right triangle. The analytical relationships among these vectors are summarized below.

A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} are defined to be the components of A size 12{A} {} along the x - and y -axes. The three vectors A size 12{A} {} , A x size 12{A rSub { size 8{x} } } {} , and A y size 12{A rSub { size 8{y} } } {} form a right triangle:

A x  + A y  = A . size 12{A rSub { size 8{x} } bold " + A" rSub { size 8{y} } bold " = A."} {}

Note that this relationship between vector components and the resultant vector holds only for vector quantities (which include both magnitude and direction). The relationship does not apply for the magnitudes alone. For example, if A x = 3 m size 12{A rSub { size 8{x} } } {} east, A y = 4 m size 12{A rSub { size 8{y} } } {} north, and A = 5 m size 12{A} {} north-east, then it is true that the vectors A x  + A y  = A size 12{A rSub { size 8{x} } bold " + A" rSub { size 8{y} } bold " = A"} {} . However, it is not true that the sum of the magnitudes of the vectors is also equal. That is,

3 m + 4 m   5 m alignl { stack { size 12{"3 M + 4 M "<>" 5 M"} {} # {}} } {}

Thus,

A x + A y A size 12{A rSub { size 8{x} } +A rSub { size 8{y} }<>A} {}

If the vector A size 12{A} {} is known, then its magnitude A size 12{A} {} (its length) and its angle θ size 12{θ} {} (its direction) are known. To find A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} , its x - and y -components, we use the following relationships for a right triangle.

A x = A cos θ size 12{A rSub { size 8{x} } =A"cos"θ} {}

and

A y = A sin θ . size 12{A rSub { size 8{y} } =A"sin"θ"."} {}
]A dotted vector A sub x whose magnitude is equal to A cosine theta is drawn from the origin along the x axis. From the head of the vector A sub x another vector A sub y whose magnitude is equal to A sine theta is drawn in the upward direction. Their resultant vector A is drawn from the tail of the vector A sub x to the head of the vector A-y at an angle theta from the x axis. Therefore vector A is the sum of the vectors A sub x and A sub y.
The magnitudes of the vector components A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} can be related to the resultant vector A size 12{A} {} and the angle θ size 12{θ} {} with trigonometric identities. Here we see that A x = A cos θ size 12{A rSub { size 8{x} } =A"cos"θ} {} and A y = A sin θ size 12{A rSub { size 8{y} } =A"sin"θ} {} .

Suppose, for example, that A size 12{A} {} is the vector representing the total displacement of the person walking in a city considered in Kinematics in Two Dimensions: An Introduction and Vector Addition and Subtraction: Graphical Methods .

In the given figure a vector A of magnitude ten point three blocks is inclined at an angle twenty nine point one degrees to the positive x axis. The horizontal component A sub x of vector A is equal to A cosine theta which is equal to ten point three blocks multiplied to cosine twenty nine point one degrees which is equal to nine blocks east. Also the vertical component A sub y of vector A is equal to A sin theta is equal to ten point three blocks multiplied to sine twenty nine point one degrees,  which is equal to five point zero blocks north.
We can use the relationships A x = A cos θ size 12{A rSub { size 8{x} } =A"cos"θ} {} and A y = A sin θ size 12{A rSub { size 8{y} } =A"sin"θ} {} to determine the magnitude of the horizontal and vertical component vectors in this example.

Questions & Answers

how vapour pressure of a liquid lost through convection
Yomzi Reply
Roofs are sometimes pushed off vertically during a tropical cyclone, and buildings sometimes explode outward when hit by a tornado. Use Bernoulli’s principle to explain these phenomena.
Aliraza Reply
Plz answer the question ☝️☝️
Aliraza
what's the basic si unit of acceleration
ELLOIN Reply
Explain why the change in velocity is different in the two frames, whereas the change in kinetic energy is the same in both.
Fabian Reply
Insulators (nonmetals) have a higher BE than metals, and it is more difficult for photons to eject electrons from insulators. Discuss how this relates to the free charges in metals that make them good conductors.
Muhammad Reply
Is the photoelectric effect a direct consequence of the wave character of EM radiation or of the particle character of EM radiation? Explain briefly.
Muhammad
Determine the total force and the absolute pressure on the bottom of a swimming pool 28.0m by 8.5m whose uniform depth is 1 .8m.
Henny Reply
how solve this problem?
Foday
P(pressure)=density ×depth×acceleration due to gravity Force =P×Area(28.0x8.5)
Fomukom
for the answer to complete, the units need specified why
muqaddas Reply
That's just how the AP grades. Otherwise, you could be talking about m/s when the answer requires m/s^2. They need to know what you are referring to.
Kyle
Suppose a speck of dust in an electrostatic precipitator has 1.0000×1012 protons in it and has a net charge of –5.00 nC (a very large charge for a small speck). How many electrons does it have?
Alexia Reply
how would I work this problem
Alexia
how can you have not an integer number of protons? If, on the other hand it supposed to be 1e12, then 1.6e-19C/proton • 1e12 protons=1.6e-7 C is the charge of the protons in the speck, so the difference between this and 5e-9C is made up by electrons
Igor
what is angular velocity
Obaapa Reply
angular velocity can be defined as the rate of change in radian over seconds.
Fidelis
Why does earth exert only a tiny downward pull?
Mya Reply
hello
Islam
Why is light bright?
Abraham Reply
what is radioactive element
Attah Reply
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
Aisha Reply
thanks so much. i undersooth well
Valdes Reply
what is physics
Nwafor Reply
is the study of matter in relation to energy
Kintu
physics can be defined as the natural science that deals with the study of motion through space,time along with its related concepts which are energy and force
Fidelis
Practice Key Terms 1

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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