Scalars and Vectors

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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.

1. William Moebs, Samuel J. Ling, Jeff Sanny. University Physics Volume 1. Openstax (Houston, Texas). Sep 19 2016. Book URL: <https://openstax.org/books/university-physics-volume-1/pages/1-introduction>. Section URL: <https://openstax.org/books/university-physics-volume-1/pages/2-1-scalars-and-vectors>.