In this explainer, we will learn how to use scale diagrams to represent the resultants of combined vector quantities and the components of vectors.
Imagine a pilot who wants to fly due north, despite taking off in a crosswind that blows steadily to the west. Represented as arrows on a diagram, the plane’s velocity and wind velocity could look like the following.
This is called a scale diagram. In scale diagrams, all the grid spaces are of equal size, and their widths and heights represent some physical quantity depending on the vectors displayed. In general, vectors of any quantity could be represented on a scale diagram: acceleration, displacement, force, and so on.
One reason scale diagrams are useful is that they let us add vectors graphically. We do this using the tip-to-tail method: placing the tip of a first vector at the tail of a second, so that the vector drawn from the tail of the first to the tip of the second is the resultant.
In the diagram above, to find the sum of the two vectors, we can translate the wind’s velocity vector so that its tail overlaps the tip of the plane’s vector, as follows.
The sum of the vectors yields a vector (in green) pointing directly north.
Note that vectors can be combined using this method even when the type of vector on a given diagram is not defined.
Example 1: Finding the Resultant of Vectors on a Scale Diagram
Some vectors are drawn to scale on a square grid. Which color vector shows the resultant of the black vectors and ?
Since all the objects shown on this diagram are arrows, we know that each one represents a vector. We can add together vectors and using the tip-to-tail method.
This will involve either sliding vector so its tail sits at the tip of vector , or moving vector so its tail is at the tip of . If we choose the second approach and shift vector , it will look as follows.
A vector from the tail of to the tip of is shown by the red vector. Our answer is option C.
Another way scale diagrams are useful is letting us solve for the components of vectors. Say that a vector is plotted on a scale diagram, as shown below.
The diagram lets us determine how many grid spaces long are the horizontal and vertical components of , as follows.
If we also know the scale to which the diagram is drawn, we can solve for the length of both components. Knowing them, we can also determine the length of the original vector, in this case .
Example 2: Solving for a Vector’s Length on a Scale Diagram
Some vectors are drawn to the scale of the ruler on a square grid. The sides of the squares are 1 cm long. The red vector is the resultant of the blue and green vectors. What is the length of the resultant vector, measured to the nearest centimetre?
In this scale diagram, we see the tip of the blue vector meeting with the tail of the green vector. This means a vector from the tail of the blue to the tip of the green is their resultant, or sum. The red vector is that resultant.
Since the green vector is purely vertical with no horizontal component, and the blue vector is purely horizontal, we can think of them as the components of the red vector.
The centimetre ruler in the diagram shows us that each grid space is one centimetre tall. Therefore, if we count the number of grid spaces the green vector is tall, we will know its height in centimetres, as shown below.
Since the grids in this example are square, each grid space is also one centimetre wide. We can use the same counting approach to find the length of the blue vector, as follows.
The green and blue vectors make up the legs of a right triangle, with the red vector being the hypotenuse.
Rather than calculating the length of the red vector though, we want to measure it using our ruler scale. We can begin to do this using a straightedge that we line up parallel to the red vector and mark at the vector’s tail and tip, as shown below.
The distance between these two marks equals the red vector’s length. We can measure that distance by turning our straightedge so it lines up parallel to the ruler scale, as follows.
The distance between the marks on our straightedge is just slightly greater than 12 cm.
To the nearest centimetre, the length of the red vector is 12 cm.
Another use of scale diagrams is allowing us to solve for the angle between a given vector and one of its components. We do this using a tool called a protractor, shown below.
This protractor has angles marked from 0 to 180 degrees, in increments of one degree.
Given an angle as the one shown above, we can measure it by positioning its vertex at the protractor’s origin (indicated here by the plus sign) and aligning one ray of the angle with the zero degree line as follows.
We can use a similar approach to measure angles in scale diagrams. Specifically, we aim to measure the angle between a vector and one of its components drawn in a scale diagram.
Example 3: Measuring a Scale Diagram Angle with a Protractor
Some vectors are drawn to scale on a square grid. The green vector is the vertical component of the red vector. The blue vector is the horizontal component of the red vector. What is the angle between the red vector and its horizontal component?
We are told in the problem statement that the green and blue vectors are components of the red vector.
We want to determine the angle between the red vector and its horizontal component, which is the blue vector. Note that our protractor is already in position so that one arm of the angle to be measured runs along the zero-degree line of the protractor and the angle vertex is at the protractor’s origin.
Therefore, we read from the zero-degree line up to where the red line crosses the finest markings.
The angle in blue, shown above, is what we seek to measure. We see that this angle is greater than 50 degrees and less than 60 degrees. To discern its measure to the nearest degree, we focus on the circled region of the protractor. Here, each line represents a difference of one degree in angle.
The red line crosses these marks two lines past 130 degrees. This is a different measurement scale than the one we use, but note that 130 degrees on this outer scale corresponds to 50 degrees on the inner scale we are using.
The angle we measure then is degrees beyond, or greater than, 50 degrees. The angle between the red vector and its horizontal component is 52 degrees.
- Scale diagrams let us represent vectors of any type on a grid.
- Because the grid spaces in a scale diagram are of equal size, we can use them to identify and measure the perpendicular components of a vector.
- Two or more vectors can be added on a scale diagram using the tip-to-tail method.
- The angle between a vector and either of its components can be measured with a protractor.