Lesson Explainer: Vector Quantities Mathematics

In this explainer, we will learn how to describe how vectors are used in mechanics to solve problems in two dimensions.

When modeling real-life motion using mathematics, we need to distinguish between vector and scalar quantities. A scalar quantity can only take positive (or zero) values that indicate the magnitudes of certain features, regardless of the direction. Distance and speed are scalar quantities because they do not specify the direction, while the corresponding real-life features have specific directions. For instance, if we are told that a plane is flying at a speed of 200 kilometres per hour, this tells us how fast it is traveling but not where it is going. Quantities like time and mass are also considered to be scalars since they do not inherently concern directions and hence can only take positive values.

On the other hand, a vector quantity indicates both the direction and magnitude. Displacement and velocity are the vector counterparts to distance and speed respectively. In other words, distance is the magnitude of displacement, and speed is the magnitude of velocity. Since an object’s velocity also indicates its direction, a plane flying north at a speed of 200 kilometres per hour will have a different velocity from another plane flying south at the same speed.

Let us consider how the direction is specified in one-dimensional (or linear) motion. In linear motion, there are exactly two different options for directions, so the directions can be specified by placing either the positive or the negative sign in front of their scalar counterparts. To achieve this, we can use the following steps.

Let us consider the velocities of the planes from the example mentioned above. Here, one plane is traveling north, while the other is traveling south at the same speed. We cannot define the velocities of the two planes without first assigning either north or south to be the positive direction. If we say that north is the positive direction, the velocity of the first plane is equal to km/h, or simply 200 km/h, while the second plane has a velocity of km/h. We note that the unit of velocity is equal to that of speed.

We note that, in the first instance, we omitted the positive sign in the velocity to write 200 km/h, which appears to be the same as its speed. In such cases, we should be careful to distinguish vector quantities from scalar quantities. Saying that a plane has a speed of 200 km/h is different from saying that it has a velocity of 200 km/h since the latter contains additional information about its direction. Even when these two quantities take the same form, they have distinct meanings. Hence, scalar and vector quantities are not directly comparable.

In our first example, we will find an object’s velocity and displacement from their scalar counterparts when their directions are specified.

Example 1: Calculating the Velocity, Speed, Distance, and Displacement along Straight Paths

The car is traveling on a highway along the route shown.

The distance between points A and B is 60 km, and the distance between points B and C is 30 km. The car travels from A to C, passing through B, then back to A, passing through B. Within each segment, the car travels at a constant speed of 90 km/h. What is the displacement of the car from C to B? Also, what is the velocity of this segment?

Answer

Displacement and velocity are vector quantities whose scalar counterparts are distance and speed respectively. Vector quantities represent direction and magnitude, while scalar quantities only indicate magnitude. We recall that in linear motion, the direction of a vector quantity can be defined purely in terms of a positive or negative sign. The given figure indicates that the positive direction is toward the right, which means that the positive sign means the right direction, while the negative sign means the left direction.

Let us first consider the displacement. The scalar counterpart of displacement is distance. Since we want to find the displacement from C to B, we need to first find the distance between these two points. We are given that the distance between B and C is 30 km, which is equal to the magnitude of the displacement. The direction going from C to B is toward the left, which is indicated by a negative sign. Hence, the displacement from C to B is given by placing a negative sign in front of the scalar counterpart, which is 30 km. This gives us km.

Next, we will find the velocity. The scalar counterpart of velocity is speed, and we are given that the car is traveling at a constant speed of 90 km/h; hence, this is equal to the magnitude of the velocity from C to B. The direction of velocity as the car travels from C to B is to the left, which is opposite the positive direction. Therefore, the velocity from C to B is given by placing a negative sign in front of the scalar counterpart, 90 km/h. This gives us km/h.

Hence, the displacement and velocity of the car on the segment from C to B is given by

In the previous example, we found an object’s displacement and velocity using their scalar counterparts. Other important vector quantities in motion are acceleration and force. Like displacement and velocity, these quantities have specific directions associated with them in motion, which is indicated by their signs in one-dimensional motion. However, their scalar counterparts do not have special names, like distance and speed, so we refer to them as the magnitude of acceleration or the magnitude of force. Both the vector and scalar counterparts share the same units. For instance, acceleration, as well as the magnitude of acceleration, can have the unit metres per second squared (m/s²), while the standard unit of force and its magnitude is newtons (N). We summarize the vector and scalar counterparts in the table below.

We recall that acceleration is the rate of change in velocity, so its interpretation is intertwined with that of the object’s velocity. While velocity indicates the direction of motion, acceleration indicates how the velocity is changing. Let us consider how to interpret the directions of velocity and acceleration in linear motion.

We can understand the properties stated above by considering the following diagram.

In the left diagram above, the velocity and acceleration are facing the same direction. For instance, we can picture a sailboat traveling to the right, where the wind, which acts as acceleration, is blowing in the direction the boat is already moving in. This would cause the sailboat to travel faster, which means that the boat’s speed would increase. In the same way, we can picture the scenario represented by the diagram on the right. In this case, the wind is blowing against the sailboat, which would slow the boat down. This leads the boat’s speed to decrease.

In our next example, we will consider this relationship between an object’s velocity and its acceleration in linear motion.

So far, we have considered vector quantities in linear motion. Direction in linear motion is simple because there are only two options for the direction. When we consider motion on a plane, that is, two-dimensional motion or planar motion, we can see that there are more than two possible directions. For instance, a car could drive toward north, south, east, or west, which gives us four directions. But these are not the only possibilities for direction in planar motion. A car could drive northwest, north-northwest, or, in fact, in any direction given by an angle starting from the north direction. This leads to an infinite number of possible directions for planar motion. Hence, we cannot hope to capture the direction in two dimensions the same way we did for one-dimensional motion.

Instead of using signs, we indicate directions in planar motion using vectors. Let us briefly review a few properties of vectors. In two dimensions, any vector can be written in terms of the standard unit vectors and . These standard unit vectors are vectors along the horizontal and vertical axes whose magnitude is equal to 1.

We can write a general two-dimensional vector starting from the origin and ending at a point by multiplying these standard unit vectors by scalars and adding them together, which leads to the expression , as shown below.

We recall that the magnitude of a vector is given by the length of the arrow, which is given by

This formula for magnitude comes from the Pythagorean theorem. We can see this in the figure above since the vector arrow (in red) forms the hypotenuse of a right triangle when combined with the horizontal and vertical components. Hence, the square of the magnitude is given by adding the squares of the lengths of the horizontal and vertical arrows, which are and respectively. This leads to the formula above for the magnitude of .

We can use vectors to represent vector quantities like displacement, velocity, acceleration, and force. The magnitudes of these vectors correspond to their scalar counterparts, which are the same as what we discussed in the context of linear motion. When we know a vector quantity of motion, we can use the Pythagorean theorem to find its scalar counterpart.

In the next example, we will find the speed of an object by finding the magnitude of the velocity.

In the previous example, we considered the magnitude of a vector quantity, which leads to its scalar counterpart. On the other hand, the direction of vector quantities can be understood using an angle in relation to either standard unit vector. We can obtain such angles by constructing an appropriate right triangle and using trigonometry, as shown below.

If we know that the vector takes the form , then we know that the green sides of the right triangle above are given by and respectively. To find the marked angle between the diagonal vector and the standard unit vector , we can use the tangent ratio .

When the angle between the vectors exceeds , then this method will need a slight modification. In such cases, we can draw a right triangle containing the acute-angle portion of this angle, leaving out a right angle in the process, as seen in the following diagram.

In the diagram above, we still end up with a right triangle, and the marked angle can be found using the tangent ratio . However, we need to add the right angle () to the marked angle to obtain the angle between and .

In our next example, we will find the angle between an object’s acceleration and a standard unit vector.

Example 4: Calculating the Direction of Acceleration

The acceleration of a particle is m/s². Find the angle, rounded to the nearest degree, that the acceleration of this particle makes with the unit vector .

Answer

Recall that acceleration is a vector quantity in motion that represents the rate of change in velocity. In this example, we are given a particle’s acceleration as a vector. Let us begin by drawing this vector and the unit vector on the same grid.

We are looking for the angle that is labeled as in the diagram above. We can see that the vertical and horizontal arrows along with the diagonal arrow form a right triangle. The two perpendicular sides have lengths 4 and 3 respectively. We should be careful not to mistake the length of the vertical side to be since the length is a scalar quantity that cannot be negative.

The side with length 4 is adjacent to angle , while the side with length 3 is opposite to . Then, the tangent ratio of this angle is given by

We can find from this equation when we apply the inverse tangent:

Rounding this answer to the nearest degree, the angle between the object’s acceleration and the unit vector is .

Usefulness of vector quantities in motion becomes apparent when we apply various vector operations. While most of these operations are reserved for future lessons, we can consider a geometric application involving vector addition. Recall that the addition of two vectors and can be represented on the Cartesian grid by connecting the two vectors so that the starting end of one vector lies on the tip of the arrow of the other vector. Then, the resulting vector can be drawn by connecting the two endpoints. This is demonstrated in the diagram below.

This geometric property of vector addition is useful in modeling real-life motion. In our final example, we will use vector addition to find the distance between two points when given displacements involving three points.

Example 5: Calculating the Magnitude of Displacement Using Vectors

A particle is moving from point to point and then to point . The displacement from point to point is cm, and the displacement from point to point is cm. Find the distance between point and point rounded to the nearest one-hundredth of a centimetre.

Answer

We recall that displacement is a vector quantity that indicates both direction and magnitude, while distance is a scalar quantity representing only magnitude. The distance we are looking for is between point and point , which is the magnitude of displacement from to . Let us first find this displacement.

Recall that the addition of two vectors can be represented on the Cartesian grid by connecting the two vectors so that the starting end of one vector lies on the tip of the arrow of the other vector. To visualize this process, let us draw a diagram containing the three points along with the two given displacements.

The bottom vector represents the displacement from to , which we are looking for. Using the geometric property of vector addition, we can see that traveling from to then from to is an equivalent displacement to traveling directly from to . In other words,

Let us compute this vector addition:

This tells us that the displacement from point to point is cm. Since distance is the magnitude of displacement, we can find the distance between these two points by calculating the magnitude of this vector. Recall that the magnitude of vector is given by

Hence, the magnitude of can be found by substituting and into the formula above:

Rounded to the nearest one-hundredth of a centimetre, the distance between point and point is 4.47 cm.

Let us finish by recapping a few important concepts from this explainer.