Describing the 3D World: Vectors

Physical phenomena take place in the 3D world around us. In order to be able to make quantitative predictions and give detailed, quantitative explanations, we need tools for describing precisely the positions and velocities of objects in 3D, and the changes in position and velocity due to interactions. These tools are mathematical entities called 3D “vectors.”

3D Coordinates

We will use a 3D coordinate system to specify positions in space and other vector quantities. Usually we will orient the axes of the coordinate system as shown in Figure 1.10: +x axis to the right, +y axis upward, and +z axis coming out of the page, toward you. This is a “right-handed” coordinate system: if you hold the thumb, first, and second fingers of your right hand perpendicular to each other, and align your thumb with the x axis and your first finger with the y axis, your second finger points along the z axis. (In some math and physics textbook discussions of 3D coordinate systems, the x axis points out, the y axis points to the right, and the z axis points up, but we will also use a 2D coordinate system with y up, so it makes sense always to have the y axis point up.)
Figure 1.10    Right-handed 3D coordinate system.

Basic Properties of Vectors: Magnitude and Direction

A vector is a quantity that has a magnitude and a direction. For example, the velocity of a baseball is a vector quantity. The magnitude of the baseball's velocity is the speed of the baseball—for example, 20 meters/second. The direction of the baseball's velocity is the direction of its motion at a particular instant—for example, “up” or “to the right” or “west” or “in the +y direction.” A symbol denoting a vector is written with an arrow over it:

Position

A position in space can also be considered to be a vector, called a position vector, pointing from an origin to that location. Figure 1.11 shows a position vector that might represent your final position if you started at the origin and walked 4 meters along the x axis, then 2 meters parallel to the z axis, then climbed a ladder so you were 3 meters above the ground. Your new position relative to the origin is a vector that can be written like this:
In three dimensions a vector is a triple of numbers . Quantities like the position of an object and the velocity of an object can be represented as vectors:
Figure 1.11    A position vector and its x, y, and z components.

Components of a Vector

Each of the numbers in the triple is referred to as a component of the vector. The x component of the vector is the number vx. The z component of the vector is . A component such as vx is not a vector, since it is only one number.

It is important to note that the x component of a vector specifies the difference between the x coordinate of the tail of the vector and the x coordinate of the tip of the vector. It does not give any information about the location of the tail of the vector (compare Figure 1.11 and Figure 1.12).
Figure 1.12    The arrow represents the vector , drawn with its tail at location .

Drawing Vectors

In Figure 1.11 we represented your position vector relative to the origin graphically by an arrow whose tail is at the origin and whose arrowhead is at your position. The length of the arrow represents the distance from the origin, and the direction of the arrow represents the direction of the vector, which is the direction of a direct path from the initial position to the final position (the “displacement”; by walking and climbing you “displaced” yourself from the origin to your final position).

Since it is difficult to draw a 3D diagram on paper, when working on paper you will usually be asked to draw vectors that all lie in a single plane. Figure 1.13 shows an arrow in the xy plane representing the vector .
Figure 1.13    The position vector , drawn at the origin, in the xy plane. The components of the vector specify the displacement from the tail to the tip. The z axis, which is not shown, comes out of the page, toward you.

Vectors and Scalars

A quantity that is represented by a single number is called a scalar. A scalar quantity does not have a direction. Examples include the mass of an object, such as 5 kg, or the temperature, such as −20°C. Vectors and scalars are very different entities; a vector can never be equal to a scalar, and a scalar cannot be added to a vector. Scalars can be positive or negative:
Although a component of a vector such as vx is not a vector, it's not a scalar either, despite being only one number. An important property of a true scalar is that its value doesn't change if we orient the xyz coordinate axes differently. Rotating the axes doesn't change an object's mass, or the temperature, but it does change what we mean by the x component of the velocity since the x axis now points in a different direction.

1.X.6  
How many numbers are needed to specify a 3D position vector?

Answer:

three
1.X.7  
How many numbers are needed to specify a scalar?

Answer:

one


Magnitude of a Vector

In Figure 1.14 we again show the vector from Figure 1.11, showing your displacement from the origin. Using a 3D extension of the Pythagorean theorem for right triangles (Figure 1.15), the net distance you have moved from the starting point is
We say that the magnitude of the position vector is
The magnitude of a vector is written either with absolute-value bars around the vector as , or simply by writing the symbol for the vector without the little arrow above it, r.
Figure 1.14    A vector representing a displacement from the origin.
Figure 1.15    The magnitude of a vector is the square root of the sum of the squares of its components (3D version of the Pythagorean theorem).

The magnitude of a vector can be calculated by taking the square root of the sum of the squares of its components (see Figure 1.15).
  Magnitude of A Vector
The magnitude of a vector is always a positive number. The magnitude of a vector is a single number, not a triple of numbers, and it is a scalar, not a vector.

The magnitude of a vector is a true scalar, because its value doesn't change if you rotate the coordinate axes. Rotating the axes changes the individual components, but the length of the arrow representing the vector doesn't change.

Can a Vector Be Positive or Negative?

question
Consider the vector . Is this vector positive? Negative? Zero?


None of these descriptions is appropriate. The x component of this vector is positive, the y component is zero, and the z component is negative. Vectors aren't positive, or negative, or zero. Their components can be positive or negative or zero, but these words just don't mean anything when used with the vector as a whole.

On the other hand, the magnitude of a vector such as is always positive.

1.X.8  
If , find .

Answer:

5.10 m
1.X.9  
Can the magnitude of a vector be a negative number?

Answer:

no


Mathematical Operations Involving Vectors

Although the algebra of vectors is similar to the scalar algebra with which you are very familiar, it is not identical. There are some algebraic operations that cannot be performed on vectors.

Algebraic operations that are legal for vectors include the following operations, which we will discuss in this chapter:

  
Adding one vector to another vector:
  
Subtracting one vector from another vector:
  
Finding the magnitude of a vector:
  
Finding a unit vector (a vector of magnitude 1):
  
Multiplying (or dividing) a vector by a scalar: or
  
Finding the rate of change of a vector: or .

In later chapters we will also see that there are two more ways of combining two vectors:

The vector dot product, whose result is a scalar
The vector cross product, whose result is a vector


Operations That Are Not Legal for Vectors

Although vector algebra is similar to the ordinary scalar algebra you have used up to now, there are certain operations that are not legal (and not meaningful) for vectors:

A vector cannot be set equal to a scalar.
A vector cannot be added to or subtracted from a scalar.
A vector cannot occur in the denominator of an expression. (Although you can't divide by a vector, note that you can legally divide by the magnitude of a vector, which is a scalar.)


1.X.10  
The vector and the scalar h = −2. What is ?

(a)  
0, −9, 1
(b)  
4, −5, 5
(c)  
4, 9, 5
(d)  
This is a meaningless expression.

Answer:

d
1.X.11  
Is 4/6, −7, 4 a meaningful expression? If so, what is its value?

Answer:

no


Multiplying a Vector by a Scalar

A vector can be multiplied (or divided) by a scalar. If a vector is multiplied by a scalar, each of the components of the vector is multiplied by the scalar:
Multiplication by a scalar “scales” a vector, keeping its direction the same but making its magnitude larger or smaller (Figure 1.16). Multiplying by a negative scalar reverses the direction of a vector.
Figure 1.16    Multiplying a vector by a scalar changes the magnitude of the vector. Multiplying by a negative scalar reverses the direction of the vector.

1.X.12  
The vector and the scalar f = −3.0. What is ?

Answer:

−0.09, 4.2, −78.0
1.X.13  
If , what is ?

Answer:

1, −1.5, 2.5


Magnitude of a Scalar

You may wonder how to find the magnitude of a quantity like , which involves the product of a scalar and a vector. This expression can be factored:
The magnitude of a scalar is its absolute value, so:

1.X.14  
If m/s, what is the magnitude of ?

Answer:

6, −9, 15 m/s
1.X.15  
How does the magnitude of the vector compare to the magnitude of the vector ?

Answer:

They are the same.


Direction of a Vector: Unit Vectors

One way to describe the direction of a vector is by specifying a unit vector. A unit vector is a vector of magnitude 1, pointing in some direction. A unit vector is written with a “hat” (caret) over it instead of an arrow. The unit vector â is called “a-hat.”
question
Is the vector a unit vector?


The magnitude of is , so this is not a unit vector.

The vector is a unit vector, since its magnitude is 1:
Note that every component of a unit vector must be less than or equal to 1.

In our 3D Cartesian coordinate system, there are three special unit vectors, oriented along the three axes. They are called i-hat, j-hat, and k-hat, and they point along the x, y, and z axes, respectively (Figure 1.17):
Figure 1.17    The unit vectors î, ĵ, .

One way to express a vector is in terms of these special unit vectors:
We will usually use the form rather than the îĵ form in this book, because the familiar notation, used in many calculus textbooks, emphasizes that a vector is a single entity.

Not all unit vectors point along an axis, as shown in Figure 1.18. For example, the vectors
are both unit vectors, since the magnitude of each is equal to 1. Note that every component of a unit vector is less than or equal to 1.
Figure 1.18    The unit vector has the same direction as the vector , but its magnitude is 1, and it has no physical units.

Calculating Unit Vectors

Any vector may be factored into the product of a unit vector in the direction of the vector, multiplied by a scalar equal to the magnitude of the vector.
For example, a vector of magnitude 5, aligned with the y axis, could be written as:
Therefore, to find a unit vector in the direction of a particular vector, we just divide the vector by its magnitude:
  Calculating A Unit Vector
example

Unit Vector

If m/s, then

Remember that to divide a vector by a scalar, you divide each component of the vector by the scalar. The result is a new vector. Note also that a unit vector has no physical units (such as meters per second), because the units in the numerator and denominator cancel.

1.X.16  
What is the unit vector in the direction of 0, 6, 0?

Answer:

0, 1, 0
1.X.17  
What is the unit vector â in the direction of , where ?

Answer:

0.873, 0.436, −0.218


Equality of Vectors

  Equality of Vectors
A vector is equal to another vector if and only if all the components of the vectors are equal.
The magnitudes and directions of two equal vectors are the same:
example

Equal Vectors

If , then


1.X.18  
If , and , what is the y component of ?

Answer:

7
1.X.19  
Consider the vectors and represented by arrows in Figure 1.19. Are these two vectors equal?
Figure 1.19    Are these two vectors equal? (Exercise 1.X.19.)

Answer:

no


Vector Addition

  Adding Vectors
The sum of two vectors is another vector, obtained by adding the components of the vectors.
example

Adding Vectors

Warning: Don't Add Magnitudes!

The magnitude of a vector is not in general equal to the sum of the magnitudes of the two original vectors! For example, the magnitude of the vector 3, 0, 0 is 3, and the magnitude of the vector −2, 0, 0 is 2, but the magnitude of the vector (3, 0, 0 + −2, 0, 0) is 1, not 5!

Adding Vectors Graphically: Tip to Tail

The sum of two vectors has a geometric interpretation. In Figure 1.20 you first walk along displacement vector , followed by walking along displacement vector . What is your net displacement vector ? The x component Cx of your net displacement is the sum of Ax and Bx. Similarly, the y component Cy of your net displacement is the sum of Ay and By.
  Graphical Addition of Vectors
To add two vectors and graphically (Figure 1.20):

  
Draw the first vector .
  
Move the second vector (without rotating it) so its tail is located at the tip of the first vector.
  
Draw a new vector from the tail of vector to the tip of vector .

Figure 1.20    The procedure for adding two vectors graphically: draw vectors tip to tail. To add graphically, move so the tail of is at the tip of , then draw a new arrow starting at the tail of and ending at the tip of .

1.X.20  
If and , what is the magnitude of ? What is the magnitude of ? What is the magnitude of ? What is the magnitude of plus the magnitude of ? Is ?

Answer:

361, 335, 577, 696, no
1.X.21  
and . Calculate the following:

(a)  
(b)  
(c)  
(d)  
(e)  

Answer:


Vector Subtraction

The difference of two vectors will be very important in this and subsequent chapters. To subtract one vector from another, we subtract the components of the second vector from the components of the first:

1.X.22  
If and , what is the sum ? What is the difference ? What is ?

Answer:


Subtracting Vectors Graphically: Tail to Tail

To subtract one vector from another vector graphically:

  
Draw the first vector .
  
Move the second vector (without rotating it) so its tail is located at the tail of the first vector.
  
Draw a new vector from the tip of vector to the tip of vector .

Note that you can check this algebraically and graphically. As shown in Figure 1.21, since the tail of is located at the tip of , then the vector should be the sum of and , as indeed it is:
Figure 1.21    The procedure for subtracting vectors graphically: draw vectors tail to tail; draw a new vector from the tip of the second vector to the tip of the first vector.

1.X.23  
Which of the following statements about the three vectors in Figure 1.22 are correct?

(a)  
(b)  
(c)  
(d)  
(e)  

Figure 1.22    Exercise 1.X.23.

Answer:

a, b, e


Commutativity and Associativity

Vector addition is commutative:
Vector subtraction is not commutative:
The associative property holds for vector addition and subtraction:

The Zero Vector

It is convenient to have a compact notation for a vector whose components are all zero. We will use the symbol to denote a zero vector, in order to distinguish it from a scalar quantity that has the value 0.
For example, the sum of two vectors .

Change in a Quantity: The Greek Letter Δ

Frequently we will want to calculate the change in a quantity. For example, we may want to know the change in a moving object's position or the change in its velocity during some time interval. The Greek letter Δ (capital delta suggesting “d for difference”) is used to denote the change in a quantity (either a scalar or a vector).

We use the subscript i to denote an initial value of a quantity, and the subscript f to denote the final value of a quantity. If a vector denotes the initial position of an object relative to the origin (its position at the beginning of a time interval), and denotes the final position of the object, then

means “change of ” or (displacement)
means “change of t” or t f − t i (time interval)

The symbol Δ (delta) always means “final minus initial, ” not “initial minus final.” For example, when a child's height changes from 1.1 m to 1.2 m, the change is Δy = +0.1 m, a positive number. If your bank account dropped from $150 to $130, what was the change in your balance? Δ (bank account) = −20 dollars.

Relative Position Vectors

Vector subtraction is used to calculate relative position vectors, vectors that represent the position of an object relative to another object. In Figure 1.23, object 1 is at location and object 2 is at location . We want the components of a vector that points from object 1 to object 2. This is the vector obtained by subtraction: . Note that the form is always “final” minus “initial” in these calculations.
Figure 1.23    Relative position vector.

1.X.24  
At 10:00 AM you are at location −3, 2, 5 m. By 10:02 AM you have walked to location 6, 4, 25 m.

(a)  
What is , the change in your position?
(b)  
What is Δt, the time interval during which your position changed?

Answer:

(a)  
9, 2, 20 m,
(b)  
120 s

1.X.25  
A snail is initially at location . At a later time the snail has crawled to location . What is , the change in the snail's position?

Answer:


Unit Vectors and Angles

Suppose that a taut string is at an angle to the +x axis, and we need a unit vector in the direction of the string. Figure 1.24 shows a unit vector pointing along the string. What is the x component of this unit vector? Consider the triangle whose base is Ax and whose hypotenuse is . From the definition of the cosine of an angle we have this:
In Figure 1.24 the angle is shown in the first quadrant (θx less than 90°), but this works for larger angles as well. For example, in Figure 1.25 the angle from the +x axis to a unit vector is in the second quadrant (θx greater than 90°) and cosθx is negative, which corresponds to Bx being negative.
Figure 1.24    A unit vector whose direction is at a known angle from the +x axis.
Figure 1.25    A unit vector in the second quadrant from the +x axis.

What is true for x is also true for y and z. Figure 1.26 shows a 3D unit vector and indicates the angles between the unit vector and the x, y, and z axes. Evidently we can write
These three cosines of the angles between a vector (or unit vector) and the coordinate axes are called the “direction cosines” of the vector. The cosine function is never greater than 1, just as no component of a unit vector can be greater than 1.
Figure 1.26    A 3D unit vector and its angles to the x, y, and z axes.

A common special case is that of a unit vector lying in the xy plane, with zero z component (Figure 1.27). In this case , so that , therefore you can express the cosine of θy as the sine of θx, which is often convenient. However, in the general 3D case shown in Figure 1.26 there is no such simple relationship among the direction angles or among their cosines.
Figure 1.27    If a vector lies in the xy plane, .

  Finding a Unit Vector from Angles
To find a unit vector if angles are given:

  
Redraw the vector of interest with its tail at the origin, and determine the angles between this vector and the axes.
  
Imagine the vector 1, 0, 0, which lies on the +x axis. θx is the angle through which you would rotate the vector 1, 0, 0 until its direction matched that of your vector. θx is positive, and .
  
θy is the angle through which you would rotate the vector 0, 1, 0 until its direction matched that of your vector. θy is positive, and .
  
θz is the angle through which you would rotate the vector 0, 0, 1 until its direction matched that of your vector. θz is positive, and .

example

From Angle to Unit Vector

A rope lying in the xy plane, pointing up and to the right, supports a climber at an angle of 20° to the vertical (Figure 1.28). What is the unit vector pointing up along the rope?
Figure 1.28    A climber supported by a rope.
Solution  
Follow the procedure given above for finding a unit vector from angles. In Figure 1.29 we redraw the vector with its tail at the origin, and we determine the angles between the vector and the axes. If we rotate the unit vector 1, 0, 0 from along the +x axis to the vector of interest, we see that we have to rotate through an angle θx = 70°. To rotate the unit vector 0, 1, 0 from along the +y axis to the vector of interest, we have to rotate through an angle of . The angle from the +z axis to our vector is . Therefore the unit vector that points along the rope is this:
Figure 1.29    Redraw the vector with its tail at the origin. Identify the angles between the positive axes and the vector. In this example the vector lies in the xy plane.


Further Discussion

You may have noticed that the y component of the unit vector can also be calculated as sin 70° = 0.940, and it is often useful to recognize that a vector component can be obtained using sine instead of cosine. There is, however, some advantage always to calculate in terms of direction cosines. This is a method that always works, including in 3D, and that avoids having to decide whether to use a sine or a cosine. Just use the cosine of the angle from the relevant positive axis to the vector.
example

From Unit Vector to Angles

A vector points from the origin to the location −600, 0, 300 m. What is the angle that this vector makes to the x axis? To the y axis? To the z axis?
Solution  
But we also know that , and the arccosine gives θx = 153.4°. Similarly,


Further Discussion

Looking down on the xz plane in Figure 1.30, you can see that the difference between θx = 153.4° and θz = 63.4° is 90°, as it should be.
Figure 1.30    Look down on the xz plane. The difference in the two angles is 90°, as it should be.

1.X.26  
A unit vector lies in the xy plane, at an angle of 160° from the +x axis, with a positive y component. What is the unit vector? (It helps to draw a diagram.)

Answer:

−0.940, 0.342, 0
1.X.27  
A string runs up and to the left in the xy plane, making an angle of 40° to the vertical. Determine the unit vector that points along the string.

Answer:

−0.643, 0.766, 0





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