Scalars and Vectors

In this chapter, Scalars and Vectors, you will learn the difference between scalar and vector quantities in physics. In addition, you will learn how to identify the magnitude and direction of a vector.

Contents

1. Scalars and Vectors

Scalars and Vectors

Many familiar physical quantities can be specified completely with a single number and an appropriate unit. [1]

A class period lasts for 50 minutes, or 90 minutes. A gas tank holds a certain number of litres (50 L, 65 L, or 10 L). Likewise, different size containers for water can hold various amounts of water (usually from 500 mL, to 1L, 2L, 4L, and up to 10L for personal storage). Distances around a track are measured in meters (40m, 50m, 100m, 200m, 400m, 800m, 1000m, 1500m).

A physical quantity that can be completely specified in this manner is called a scalar quantity — a synonym for “number” that is measured in some known unit. We attach a number to some unit — a unit of Time, Mass, Distance, Length, Volume, Temperature, Amount, and Energy are all examples of scalar quantities. [1]

Scalar quantities that have the same physical units can be added or subtracted to one another. For scalars the normal rules of algebra apply.

Using the example of a running track, common distances ran on the track are 100m, 200, 400m (one full revolution), 800m, 1000m, 1200m, 1500m, and 1600m (1 Mile). If you run 400m, you have completed one complete revolution of the track. If you do this twice, you have 800m, if you do this 2.5x you have 1km. If you complete 4 revolutions, you get about 1 Mile (1600m).

You can add the distances in each revolution (400+400+400+400=1600m) in the four times you go around to get your total distance. If you are a distance runner, you can simply count the number of times you go around (10 rounds = 10 \times 400m = 4000m = 4km, 20 rounds = 8000m = 8k, 25 rounds = 10km) and so on.

You can add the distance in each round to get the total number of rounds. You can also multiple and divide scalars by a number to get another number, as we did above. When we multiply/divide the scalar by a number, we get the same scalar quantity (specified by a unit) but larger/smaller. [1]

If we run 4x around the track we travel 4(400m)=1600m. If we have a protein bars with 200 kcal each, and we eat ten of them, and each also has 50g of protein, then we consume 10(200kcal)=2000kcal. Of which about 500g (=10(50g)) is protein.

For example, if yesterday you at your normal calorie average (lets say 2000 kcal) and today you doubled it (perhaps because you ran that 8km on the track, so your new calorie total for the day is 2(2000 kcal = 4000 kcal) — which oddly enough might be reasonable if you ran 10km that day.

If yesterday you ate a meal that was 200 kcal of energy, and today you ate one 4x larger, 4(200 kcal) = 800 kcal. [1]

A class ending 10 min earlier than 50 min – 10 min = 40 min. [1] If you eat a 60 kcal salad, followed by two 200 kcal donuts, you eat a total of (60 + 200 + 200 = 460 kcal) of energy.

Two scalar quantities with different units can also be multiplied by one another to form a derived scalar quantity (or a derived unit). For example, if a runner covers 40 km in 4 hours, he is travelling at 10 km/h on average. In this case we have created speed as a derived scalar quantity obtained by dividing distance/time.

There are many physical quantities that cannot be described with a single number. If you need directions to a town, directions to a meeting place, of if Search and Rescue (SAR) are sent out for people in distress, they need more than one number.

If the U.S. Coast Guard dispatches a ship/helicopter for a rescue mission, then they need two unique numbers to guide them to where the rescue team needs to go. The obvious answer is distance and direction; how far away the distress call is, and in what direction. Interestingly, we can also get those two values from a different type of unit, a GPS Coordinate (which is itself a derived scalar quantity binding two units of length).

Physical quantities that can be specified completely by giving both a number of units (magnitude) and direction, they are called vector quantities.

Examples of vector quantities include displacement, velocity, position, force, and torque. [1] In the language of mathematics, physical vector quantities are represented by mathematical objects called vectors. [1] We can add and subtract two vectors, multiply a vector by a scalar or a smaller vector. [1] However, we cannot divide a vector. The operation of division by a vector is not defined. [1]

Vector Algebra Using A Graphical Method

Lets imagine vector algebra using a graphical method to be aware of basic terms and to develop a qualitative understanding.

In practice, when it comes to solve problems analytical methods are used. (These will be seen in the next section.) These are more simple computationally, and more accurate than graphical methods.

From now on, distinguish between a vector and scalar using the common convention: a letter in bold with an arrow on top denotes a vector, while a letter that is not bold without an error denotes a scalar.

A distance of 2.0 km is a scalar quantity, $d=2.0km$. The displacement of an object in its coordinate system, is referenced by a direction as well, and is a vector quantity, denoted by $\mathbf{\vec{d}} = 2.0km$.

If you tell a friend on a camping trip, where he should go in order to meet you, or give him directions to an interesting location, you cannot just tell him how far away it is. You also have to tell him what direction to go in. Otherwise, the chances of them being able to reach either location, are minimal.

You can say something like, I am located at [blank]. Or, from my location, if you travel 7.4 kilometers towards the bearing of 217 degrees Northeast, you will find a (waterfall/hut/bridge/lake).

The key concept here is that you need two pieces of information. A magnitude and a direction. Or in the case of the GPS coordinates I gave for my location, you need a longitude and latitude. In both cases, two pieces of information are required.

Displacement, a vector quantity, is a general term that we use to describe a change in position. If we walk from the tent (Point A) to the fishing hole (Point B) as we can see in the diagram below, the vector $\vec{\mathbf{D}}$, represents our displacement. This is the arrow drawn from Location A to Location B.

The length of the arrow represents its magnitude, $D$, of vector $\vec{\mathbf{D}}$. Let us say that the magnitude of this vector is $D=9km$. The magnitude of the vector is it’s length, always a positive number, so we can also signify the “magnitude” using the absolute value notation placed around our vector symbol: $D \equates | \vec{\mathbf{D}} |$.

To solve a vector problem graphically, we have to draw it to scale. The most obvious example of this is navigation using maps, which are drawn to scale. Any vector (arrow) drawn on a map immediately has a magnitude associated with it because of the scale of the map.

If we create a scale for a diagram, and decide that one unit of distance (1 km) is equal to a line segment with length $u=2cm$. To draw a vector that is 9km long according to our scale, calculate the length of the vector as $d=9u=9(2cm)=18cm$. [1]

We can write $D=9km$ to denote the magnitude of the actual displacement, and d=18cm to denote the length of the vector in drawing.

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Suppose you walk from the campsite (Point A) to the fishing pond (Point B), and then back. The magnitude of the displacement vector $\vec{\mathbf{D}}_{AB}$ is equal to the magnitude of displacement vector $\vec{\mathbf{D}}_{BA}$ (the return trip back).

We can say that $D_{AB} = D_{BA}$ since the distance travelled is the same. In both cases it is 9km. However, these vectors have two different directions, and even while the same magnitude, are not equal to one another. Ultimately, they cancel out, $\vec{\mathbf{D}}_{Total}=0$.

Because the vectors have two different directions, $\vec{\mathbf{D}}_{AB} \neq \vec{\mathbf{D}}_{BA}$.

We see that the first trip, from the campsite to the fishing hole, $\vec{\mathbf{D}}_{AB}$, travels northeast from $A$ to $B$. The return trip, $\vec{\mathbf{D}}_{BA}$, pointing southwest exactly 180 degrees in the opposite direction. [1]

We say that $\vec{\mathbf{D}}_{AB}$ is antiparallel to $\vec{\mathbf{D}}_{BA}$, and write $\vec{\mathbf{D}}_{AB} = \minus \vec{\mathbf{D}}_{BA}$.

Two vectors with an identical direction are called parallel vectors. Two parallel vectors $\vec{\mathbf{A}}$ and $\vec{\mathbf{B}}$, are equal, $\vec{\mathbf{A}}=\vec{\mathbf{B}}$, only in the case where they have equal magnitudes as well $ | \vec{\mathbf{A}} | = | \vec{\mathbf{B}} | $.

Two vectors at a perpendicular direction to one another are called orthogonal vectors. These relations between vectors — parallel, antiparallel, and orthogonal — are described in the image below.

Algebra of Vectors In One Dimension

Vectors can be multiplied by scalars, added or subtracted from other vectors.

Imagine that there is a lookout point at three-quarters the distance to the lake, $0.75D_{AB} = 4.5 km$, at point $C$. This new vector is parallel to the original vector. We can write this in the vector equation as

\[ \vec{\mathbf{D}}_{AC} = 0.75 \vec{\mathbf{D}}_{AB} \]

In a vector equation both sides of the equation are vectors. In this example we multiplied a vector by a positive scalar $\alpha = 0.75$ which created a new vector parallel to the original vector.

When a vector $\vec{\mathbf{A}}$ is multipled by a positive scalar, the result is a new vector $\vec{\mathbf{B}}$ parallel to $\vec{\mathbf{A}}$:

\[ \vec{\mathbf{B}} = \alpha \vec{\mathbf{A}} \]

The magnitude of the new vector $|\vec{\mathbf{B}}|$ is obtained by multiplying the magnitude of the original vector$|\vec{\mathbf{A}}|$ by the scalar, as expressed in the following scalar equation:

\[ B = | \alpha | A \]

Notes

[Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction]

2.1 Scalars and Vectors

Learning Objectives

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

• Describe the difference between vector and scalar quantities.
• Identify the magnitude and direction of a vector.
• Explain the effect of multiplying a vector quantity by a scalar.
• Describe how one-dimensional vector quantities are added or subtracted.
• Explain the geometric construction for the addition or subtraction of vectors in a plane.
• Distinguish between a vector equation and a scalar equation.

Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, “a class period lasts 50 min” or “the gas tank in my car holds 65 L” or “the distance between two posts is 100 m.” A physical quantity that can be specified completely in this manner is called a scalar quantity. Scalar is a synonym of “number.” Time, mass, distance, length, volume, temperature, and energy are examples of scalar quantities.

Scalar quantities that have the same physical units can be added or subtracted according to the usual rules of algebra for numbers. For example, a class ending 10 min earlier than 50 min lasts 50min−10min=40min. Similarly, a 60-cal serving of corn followed by a 200-cal serving of donuts gives 60cal+200cal=260cal of energy. When we multiply a scalar quantity by a number, we obtain the same scalar quantity but with a larger (or smaller) value. For example, if yesterday’s breakfast had 200 cal of energy and today’s breakfast has four times as much energy as it had yesterday, then today’s breakfast has 4⁢(200cal)=800cal of energy. Two scalar quantities can also be multiplied or divided by each other to form a derived scalar quantity. For example, if a train covers a distance of 100 km in 1.0 h, its speed is 100.0 km/1.0 h = 27.8 m/s, where the speed is a derived scalar quantity obtained by dividing distance by time.

Many physical quantities, however, cannot be described completely by just a single number of physical units. For example, when the U.S. Coast Guard dispatches a ship or a helicopter for a rescue mission, the rescue team must know not only the distance to the distress signal, but also the direction from which the signal is coming so they can get to its origin as quickly as possible. Physical quantities specified completely by giving a number of units (magnitude) and a direction are called vector quantities. Examples of vector quantities include displacement, velocity, position, force, and torque. In the language of mathematics, physical vector quantities are represented by mathematical objects called vectors (Figure 2.2). We can add or subtract two vectors, and we can multiply a vector by a scalar or by another vector, but we cannot divide by a vector. The operation of division by a vector is not defined.

Figure 2.2 We draw a vector from the initial point or origin (called the “tail” of a vector) to the end or terminal point (called the “head” of a vector), marked by an arrowhead. Magnitude is the length of a vector and is always a positive scalar quantity. (credit “photo”: modification of work by Cate Sevilla)

Let’s examine vector algebra using a graphical method to be aware of basic terms and to develop a qualitative understanding. In practice, however, when it comes to solving physics problems, we use analytical methods, which we’ll see in the next section. Analytical methods are more simple computationally and more accurate than graphical methods. From now on, to distinguish between a vector and a scalar quantity, we adopt the common convention that a letter in bold type with an arrow above it denotes a vector, and a letter without an arrow denotes a scalar. For example, a distance of 2.0 km, which is a scalar quantity, is denoted by d = 2.0 km, whereas a displacement of 2.0 km in some direction, which is a vector quantity, is denoted by →𝐝.

Suppose you tell a friend on a camping trip that you have discovered a terrific fishing hole 6 km from your tent. It is unlikely your friend would be able to find the hole easily unless you also communicate the direction in which it can be found with respect to your campsite. You may say, for example, “Walk about 6 km northeast from my tent.” The key concept here is that you have to give not one but two pieces of information—namely, the distance or magnitude (6 km) and the direction (northeast).

Displacement is a general term used to describe a change in position, such as during a trip from the tent to the fishing hole. Displacement is an example of a vector quantity. If you walk from the tent (location A) to the hole (location B), as shown in Figure 2.3, the vector →𝐃, representing your displacement, is drawn as the arrow that originates at point A and ends at point B. The arrowhead marks the end of the vector. The direction of the displacement vector →𝐃 is the direction of the arrow. The length of the arrow represents the magnitude D of vector →𝐃. Here, D = 6 km. Since the magnitude of a vector is its length, which is a positive number, the magnitude is also indicated by placing the absolute value notation around the symbol that denotes the vector; so, we can write equivalently that 𝐷≡∣→𝐃∣. To solve a vector problem graphically, we need to draw the vector →𝐃 to scale. For example, if we assume 1 unit of distance (1 km) is represented in the drawing by a line segment of length u = 2 cm, then the total displacement in this example is represented by a vector of length 𝑑=6⁢𝑢=6⁢(2cm)=12cm, as shown in Figure 2.4. Notice that here, to avoid confusion, we used 𝐷=6km to denote the magnitude of the actual displacement and d = 12 cm to denote the length of its representation in the drawing.

Figure 2.3 The displacement vector from point A (the initial position at the campsite) to point B (the final position at the fishing hole) is indicated by an arrow with origin at point A and end at point B. The displacement is the same for any of the actual paths (dashed curves) that may be taken between points A and B.

Figure 2.4 A displacement →𝐃 of magnitude 6 km is drawn to scale as a vector of length 12 cm when the length of 2 cm represents 1 unit of displacement (which in this case is 1 km).

Suppose your friend walks from the campsite at A to the fishing pond at B and then walks back: from the fishing pond at B to the campsite at A. The magnitude of the displacement vector →𝐃𝐴⁢𝐵 from A to B is the same as the magnitude of the displacement vector →𝐃𝐵⁢𝐴 from B to A (it equals 6 km in both cases), so we can write 𝐷𝐴⁢𝐵=𝐷𝐵⁢𝐴. However, vector →𝐃𝐴⁢𝐵 is not equal to vector →𝐃𝐵⁢𝐴 because these two vectors have different directions: →𝐃𝐴⁢𝐵≠→𝐃𝐵⁢𝐴. In Figure 2.3, vector →𝐃𝐵⁢𝐴 would be represented by a vector with an origin at point B and an end at point A, indicating vector →𝐃𝐵⁢𝐴 points to the southwest, which is exactly 180° opposite to the direction of vector →𝐃𝐴⁢𝐵. We say that vector →𝐃𝐵⁢𝐴 is antiparallel to vector →𝐃𝐴⁢𝐵 and write →𝐃𝐴⁢𝐵=−→𝐃𝐵⁢𝐴, where the minus sign indicates the antiparallel direction.

Two vectors that have identical directions are said to be parallel vectors—meaning, they are parallel to each other. Two parallel vectors →𝐀 and →𝐁 are equal, denoted by →𝐀=→𝐁, if and only if they have equal magnitudes ∣→𝐀∣=∣→𝐁∣. Two vectors with directions perpendicular to each other are said to be orthogonal vectors. These relations between vectors are illustrated in Figure 2.5.

Figure 2.5 Various relations between two vectors →𝐀 and →𝐁. (a) →𝐀≠→𝐁 because 𝐴≠𝐵. (b) →𝐀≠→𝐁 because they are not parallel and 𝐴≠𝐵. (c) →𝐀≠−→𝐀 because they have different directions (even though ∣→𝐀∣=∣−→𝐀∣=𝐴). (d) →𝐀=→𝐁 because they are parallel and have identical magnitudes A = B. (e) →𝐀≠→𝐁 because they have different directions (are not parallel); here, their directions differ by 90°—meaning, they are orthogonal.

Check Your Understanding 2.1

Two motorboats named Alice and Bob are moving on a lake. Given the information about their velocity vectors in each of the following situations, indicate whether their velocity vectors are equal or otherwise. (a) Alice moves north at 6 knots and Bob moves west at 6 knots. (b) Alice moves west at 6 knots and Bob moves west at 3 knots. (c) Alice moves northeast at 6 knots and Bob moves south at 3 knots. (d) Alice moves northeast at 6 knots and Bob moves southwest at 6 knots. (e) Alice moves northeast at 2 knots and Bob moves closer to the shore northeast at 2 knots.

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