Chapter 2 Vectors

We were limited in our discussion of motion in Chapter 1. From now on we will discuss motion in more than 1 direction (dimension). Most of our work with motion will involve two dimensions. i.e. We will often use a Cartesian coordinate system with X and Y axes.

page 2-2 Vectors

Many quantities in the real world are best described using vectors. In this chapter "displacement" refers to the change in location, from point A to point B, not to the pushing aside of a liquid or gas. The math techniques we will need for vectors are not too far removed from the algebra and trigonometry you already are familiar with.

Displacement Vectors

page 2-3 *What two quantities must be specified to describe a displacement vector?

Why are the displacement vectors shown in Figure 1 the same?

In what way is the magnitude of a vector equal to the size of the vector?

 

Just for some contrast, let me define a scalar quantity. A scalar is specified by giving the size of the quantity (along with the units). Scalars do not have an associated direction. e.g. mass, time, energy.

Can you list some quantities that are vectors other than the displacement vector?

 

The methods involved in solving physics problems that involve scalars are different than the methods involved in solving problems that involve vectors. You must learn to recognize whether the problem has a scalar nature or a vector nature. In most cases in this book, vector problems require you to work 2 problems simultaneously. One calculation will be in the X direction and the other calculation will be in the Y direction. Often the X and Y results will be combined through the use of right-triangle calculations.

Arithmetic of Vectors

Is the net displacement equal to the addition of the lengths of the two vectors in Figure 2?

What is different about the meaning of "+" in 405 + 190 and A + B ?

(A bold and underlined letter represents a vector.)

 

The addition of vector quantities does not follow the ordinary rules of algebra. You can only perform ordinary addition or subtraction if the vectors are parallel to each other. Vectors can be added graphically to obtain an approximate result. Vectors can be added mathematically to obtain an exact result. The sum of two vectors is called the net or resultant.

Under this paragraph, make a small Cartesian coordinate system and label the upward line, North. What is the approximate addition of a vector of 5 units to the east, A , and a vector that has a length of 7 units in the northwest direction, B? Draw A by starting the arrow at the origin of your coordinate system. If you understand Figure 1 you will understand that you can draw many vectors that are equal to B. Draw a vector that is equal to B by starting the vector at the arrow end of A . Draw the net vector from the origin to the arrow end of B.

 

 

 

 

 

 

Draw a second small Cartesian coordinate system. This time draw B first. Then start A at the tip of B . Draw the net vector. Compare the two net vectors.

 

page 2-4 Vector addition does posses the property called commutation. That is, you can start the sum process with any of the vectors that need to be added and still obtain the same result. A + B is equal to B + A . Vector addition also obeys the associative law as shown in the text.

Vector Subtraction and Multiplication of a Vector by a Scalar

Vector subtraction A – B is accomplished by adding the negative of the B vector. The negative of a vector has the same magnitude but a direction 180 degrees different than the positive version of the vector. On a graph you simply draw the arrow at the other end of a vector that is parallel and the same length as the original vector.

page 2-5 Multiplication of a Vector by a Number

Scalar multiplication of vectors involves multiplying a vector by a scalar, e.g. 3A. Scalar multiplication affects the magnitude but not the direction of the vector. If A has a magnitude of 5, 3A has a magnitude of 15. i.e. 3A = A + A + A.

What questions do you have on pages 2:4-5?

page 2-6 Magnitude of a Vector

Graphical Work

I don’t require you to use a ruler or protractor as you make vector drawings. Just try to be reasonably careful with the magnitude and direction on your graph.

 

 

pages 2:8-10 Adding Vectors by Components

To perform the mathematical addition of vectors we must be able to resolve vectors into their components. This is accomplished by the trigonometric calculations for a right triangle. The vector is always the hypotenuse in the right triangle. The X and Y components of the vector are the shorter sides of the right triangle. If the angle is given in standard form (i.e. measured counterclockwise from the positive X axis) then you can always use the relationships below to find the components of the vector. If A is the magnitude of the vector then

AX = A cos q and AY = A sin q .

Is AX a scalar or a vector?

Suppose a certain vector has a magnitude of 12 meters and a direction of 37 degrees (measured in the usual sense from the positive X axis). What are the values of the components of the vector?

AX Ay

 

If you are given the components of a vector you can reconstruct the vector using the Pythagorean Theorem and the tangent function (plus thinking).

A = sqrt(A2X + A2Y ) tan q = AY /AX

*How is q found from this relationship?

Is the result your calculator displays for q always the correct angle?

Perform the reconstruction of A using the AX and AY values you found above.

Are you surprised by the result?

 

Mathematical addition of vectors ( C = A + B ) is performed by 1) finding the components of each vector A and B, 2) adding "like" components e.g. CX = AX + BX and CY = AY + BY , 3) reconstructing C.

Let B have a length of 7 meters and an angle of 160 degrees. Find C = A + B by using the component method. Use the A given above. I want you to find the magnitude and direction for C .

 

 

 

 

 

 

page 2-11 Vector Multiplication

Vector multiplication is more complex than multiplication of numbers. This is to be expected since a vector is more complicated than a number. A vector has direction information. How should we "multiply" the direction information? We will use two types of vector multiplication, the "dot" product and the "cross" product. Both types of multiplications are used in physics to allow you to solve certain types of problems.

The dot product produces a result that is a scalar. The cross product produces a result that is a vector.

page 2-11b is nice background reading but is beyond the scope of this course.

page 2-12 Scalar or Dot Product

This page lays a good foundation for the dot product but we will not be calculating A dot A

We will use A dot B = AXBX + AYBY and A dot B = AB cos θ .

Calculate A dot B using the definitions of the vectors on the previous page. Use both methods of calculating the dot product. Do your results agree?

 

 

 

 

 

 

page 2-14 Interpretation of the Dot Product

*What is the condition necessary to produce the maximum result for the dot product of two vectors? (describe something about the directions of the two vectors)

What are the three conditions that will cause the dot product of two vectors to have a value of zero?

1. 2. 3.

 

*What is the condition necessary such that the dot product of the two vectors is a negative number?

 

Physical Use of the Dot Product

What questions do you have on this section?

page 2-15 Vector Cross Product We will use the definitions of A and B given on page 3. You should lay out a pen or pencil on a table in front of you to represent A. Lay out a second, shorter, pen or pencil to represent B.

A cross B produces a vector that is perpendicular to both A and B. A and B define a plane (in this case the XY plane). How many vectors are perpendicular to both A and B?

The Right Hand Rule has three steps. 1. Point the fingers of your right hand in the direction of the first vector listed in the cross product. 2. If necessary rotate your wrist such that it is easy to bend your fingers into the direction of the second vector listed in the cross product. (Hint: If you just bent your fingers towards the back of your hand and you are in great pain, you did not bend your fingers in the correct direction. Rotate your wrist. Physics is not supposed to be painful! ) 3) Your thumb (on your right hand) is now pointing in the direction of the vector result for this cross product. The calculation of the magnitude of the vector result is described below.

What questions do you have on page 2-15?

page 2-17 Magnitude of the Cross Product Method 1: A cross B = AB sin θ

Calculate the magnitude of A cross B .

If you are working on a table top, what is the direction of A cross B ?

*What is the condition necessary to produce the maximum result for the cross product of two vectors? (describe something about the directions of the two vectors)

What are the three conditions that will cause the cross product of two vectors to have a value of zero?

1. 2. 3.

Component Formula for the Cross Product

I am going to introduce "unit vectors" at this point in the course. A unit vector has a magnitude of 1. There are three unit vectors that will be important for us. i is the unit vector directed along the positive X axis. j is the unit vector directed along the positive Y axis. k is the unit vector directed along the positive Z axis.

Suppose a certain vector D has these components: D X = 4 D Y = 3 D Z = 8. We could write D = 4i + 3j + 8k .

Method 2: A cross B = i j k The method is completed by

AX AY AZ calculating the determinant. I will

BX BY BZ show the determinant steps in class.

pages 2-19 and following. Please don’t tear out the pages.

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