No products in the cart.
Scalar measurements, such as time, mass, speed, and distance have numerical values and a unit of measurement. For example:
Vector quantities, such as displacement, velocity, and acceleration, include numerical values, a unit of measurement and a direction. For example:
Notice that variables representing vectors have arrows above them. Whenever you specify a vector quantity using a value and a direction, you add the vector symbol (arrow) above the variable.
Vectors in 2-Dimensions
Notebook
Suggested Answer
Describing Vectors
A vector is represented as an arrow, with a “head” or “tip” and a “tail,” as shown below. The arrowhead shows the vector’s direction.
Representing a Vector Graphically and Numerically
For example, consider this displacement vector represented graphically beside a reference point. In this course the reference point is often at the center of a 4-point compass rose indicating north, south east, and west.
This displacement vector may be shown graphically, or may be written numerically as:
8 m [E] or 8 m [east].
All vectors have a direction but not all vectors point directly north, south, east or west, so…
How would you communicate a vector’s direction?
There are several acceptable ways to communicate a vector’s direction in two-dimensions.
In the diagram shown below, a vector points 50° east of north. In other words, the direction of the vector would be start north then rotate 50° towards the east.
Instead of drawing the vector, you can also state the vector using written notation.
In this example, you would write [N50°E]. The brackets show that it is vector notation. The above example used North, East, South and West as labels for direction of the vector but, other direction labels are also acceptable.
Let’s look at few examples:
Example 1
A velocity of 80 km/h [NW] or [N45°W]
Example 2
An acceleration of 5 m/s² [S25°E] or [E65°S]
Self-Check and Reflection
This is a self-check, which will help you to:
- assess and evaluate your own work
- determine where you are in your learning, where you need to go, and how best to get there
You will be provided with suggested answers to compare to your own responses.
After checking your responses, ask yourself the following reflection questions:
- What concepts do I need to improve upon?
- What did I do well?
- What are my next steps to ensure I understand all the concepts?
- What steps might I take to improve and grow as a learner?
State the direction of each of the three vectors shown below.
Vector 1:
Suggested Answer
[SW] or [S45°W] or [W45°S]
Vector 2:
Suggested Answer
[E 25° N] or [N 65° E]
Vector 3:
Suggested Answer
[N 15° W] or [W 75° N]
Practice: Representing Vectors
In a vector diagram, the magnitude of a vector quantity is represented by the size of the vector arrow. If the size of the arrow in each consecutive frame of the vector diagram is the same, then the magnitude of that vector is constant. For this and future questions, copy the question and your answer into your notebook for future reference, then check your response against the suggested answer.
Question 1:
Review each vector diagram on the left and identify the correct direction on the right.
Vector Diagram
Direction
Answers
Vector Diagram
Direction
[W 50° S]
[E 8° N]
[S 50° W]
[N 8° E]
For the next series of Multiple-Choice Questions, study each vector diagram shown and choose the correct direction.
Question 1 of 3
This is an image of:
Answer
[S 45° W]
Question 2 of 3
This is an image of:
Answer
[E 25° N]
Question 3 of 3
This is an image of:
Answer
Drawing Vectors
There are situations in which it is useful to draw scale diagrams of vectors. When you do this, always include a statement such as
“Let 1 cm = 2 m” to communicate the size of the vector being drawn. For example, to draw a vector that is 10 m long, you may draw a vector that is 5 cm long on your paper and include a statement that “1 cm = 2 m.”
Notebook
You will need a ruler, a protractor and compass to complete the work in this section.
Consider the velocity vectors that were mentioned previously:
These velocities can be represented with vectors drawn to scale, as shown below.
Choose a scale that is easy to draw. To find the length of line you need to draw, take the original measurement and divide by the scale.
To draw vectors that do not lie exactly along a compass direction (N, S, E, or W), it is necessary to work from a reference point with its own compass directions.
Notebook
In your Notebook, follow these directions to draw vectors when the direction includes an angle.
Draw a reference point.
Use your protractor to find the position of the angle.
Example: If the angle is [east 30° north] start on the east line and move 30° north.
Choose a scale.
Example: If the vector is 200 km, choose a scale of 1 cm = 100 km. The length of the line should be 2 cm.
Draw the vector along the angle to the correct length (measured using the scale).
Practice: Representing Vector Directions
Notebook
For each of the following questions, state the direction of the vector shown using the indicated angle.
Vector 1
Vector 2
Vector 3
Vector 4
Answer
Vector 1:
Vector 2:
Vector 3:
Vector 4:
Vector Addition
So far, we’ve learned that vectors consist of two parts, magnitude and direction. Breaking down a vector into horizontal and vertical components is a very useful technique in understanding physics problems.
Think about the following activities and how vectors are involved:
Rowing a boat across a river.
Flying a plane when there is a wind.
Whenever you see motion at an angle, you should think of it as moving horizontally and vertically at the same time. Simplifying vectors in this way can speed calculations and help to keep track of the motion of objects.
Components of a Vector
The horizontal component stretches from the start of the vector to its furthest x-coordinate. The vertical component stretches from the x-axis to the most vertical point on the vector. Together, the two components and the vector form a right triangle.
Watch this video of Mr. Andersen as he gives us a recap of the differences between scalar and vectors quantities. He also uses a demonstration to show the importance of vectors and vector addition which starts at 4:46 second mark of the video
By taking the vector to be analyzed as the hypotenuse, the horizontal and vertical components can be found by completing a right triangle. The bottom edge of the triangle is the horizontal component and the side opposite the angle is the vertical component.
Before we can really dive into the addition of vectors, let’s refresh our math memories …
right-angle triangle
a² + b² = c²
cosine
hypotenuse
sine
tangent
Review of the Pythagorean Triangle and Trigonometry
Watch the following videos to review the mathematical concepts of the Pythagorean theorem and right-angle trigonometry before continuing with this section.
Watch This!
These video links have been provided as suggestions only. You are encouraged to search for additional resources to support your understanding
When it comes to adding (and subtracting) vectors there are three methods:
- Method 1: Using a scale diagram
- Method 2: Using algebra – trigonometry (sine and cosine law)
- Method 3: Using perpendicular components
Although in this unit you are learning about vectors of motion (e.g. displacement, velocity and acceleration), you will also be applying vector concepts throughout this course. Knowing how to draw and represent a vector in physics is an essential skill for this course. Let’s see how we apply this skill next.
Method 1: Vector Addition Using Scale Diagrams
When adding vectors, you will be drawing the first vector and then placing the “tail” of the next vector against the “head” or “tip” of the first vector drawn. This is called the “tail-to-tip” or “tail-to-head” method.
The resultant vector is found by connecting the “tail” of the first vector to the “head” or “tip” of the last vector. In other words, if you are finding the resultant displacement by adding two displacement vectors, you would be connecting the “Start” position to the “Finish” position with a vector.
Still not sure how to use this 1st method? Watch this Crash Course Video(Open in new window).
This video link has been provided as a suggestion only. You are encouraged to search for additional resources to support your understanding.
Example 1:
This is Ryan. He’s listening to his favorite playlist on his way to meet friends.
Find the displacement for Ryan when he walks 14 m [N] and then 25 m [S].
Solution:
Using a vector diagram, draw both displacement vectors, plus the resultant.
The tail of vector 2 was placed alongside the head of vector 1. The resultant (or total) displacement is found by connecting the “Start” position to the “Finish” position with a vector. The vectors must be drawn with arrowheads to indicate their direction.
But what happens when vectors are not aligned in one dimension?
Example 2:
Felicia is on her way to meet Ryan, however she’s on the other side of town. She walked 25 m [east] and then 30 m [north]. Find her displacement.
Suggested Answer
Step 1:
Step 2:
Step 3:
Self-Check
OK, let’s test your knowledge on the concepts we’ve covered so far before we move on to Methods 2 and 3.
There is a certain order to drawing a scale diagram when solving for displacement. What is the correct order?
Using the menus, put the steps in the correct order for finding the displacement.
Step 1:
Draw the two displacement vectors, connecting them tip to tail.
Measure the angle between the tails of the vectors.
Measure the length of the resultant vector.
Choose a scale.
Use the scale to change the length of the resultant vector into a displacement.
Draw the resultant vector from the start to the finish on your diagram.
Step 2:
Draw the two displacement vectors, connecting them tip to tail.
Measure the angle between the tails of the vectors.
Measure the length of the resultant vector.
Choose a scale.
Use the scale to change the length of the resultant vector into a displacement.
Draw the resultant vector from the start to the finish on your diagram.
Step 3:
Draw the two displacement vectors, connecting them tip to tail.
Measure the angle between the tails of the vectors.
Measure the length of the resultant vector.
Choose a scale.
Use the scale to change the length of the resultant vector into a displacement.
Draw the resultant vector from the start to the finish on your diagram.
Step 4:
Draw the two displacement vectors, connecting them tip to tail.
Measure the angle between the tails of the vectors.
Measure the length of the resultant vector.
Choose a scale.
Use the scale to change the length of the resultant vector into a displacement.
Draw the resultant vector from the start to the finish on your diagram.
Step 5:
Draw the two displacement vectors, connecting them tip to tail.
Measure the angle between the tails of the vectors.
Measure the length of the resultant vector.
Choose a scale.
Use the scale to change the length of the resultant vector into a displacement.
Draw the resultant vector from the start to the finish on your diagram.
Step 6:
Draw the two displacement vectors, connecting them tip to tail.
Measure the angle between the tails of the vectors.
Measure the length of the resultant vector.
Choose a scale.
Use the scale to change the length of the resultant vector into a displacement.
Draw the resultant vector from the start to the finish on your diagram.
Answer
Step 1:
Choose a scale.
Step 2:
Draw the two displacement vectors, connecting them tip to tail.
Step 3:
Draw the resultant vector from the start to the finish on your diagram.
Step 4:
Measure the length of the resultant vector.
Step 5:
Use the scale to change the length of the resultant vector into a displacement.
Step 6:
Measure the angle between the tails of the vectors.
Notebook
Ready for another one?
Use your ruler and protractor to draw a vector diagram to solve this problem.
A truckload of computers travels across Ontario 45 km [west] and then 65 km [south].
Find the truck’s displacement.
Suggested Answer
Drawing the scale diagram can sometimes give you an answer slightly different from the correct answer because it is hard to measure everything exactly.
A more exact method for finding the answer uses algebra instead of scale diagrams. Next, we will learn more about using algebra to add vectors. → Method 2: Vector Addition Using Algebra.
Method 2: Vector Addition Using Algebra
FYI: you may want to review the basic concepts of trigonometry that you need to better understand this learning activity.
Let’s go back to the first few examples from this topic. But this time we’ll solve it again using algebra techniques instead of scale diagrams. The answer should be the same.
Example 1: Remember Ryan? Find the displacement for Ryan when he walks 14 m [N] and then 25 m [S].
Suggested Answer
Example 2: Remember Felicia? She walked 25 m [east] and then 30.0 m [north]. Find her displacement.
Suggested Answer
Solution: Vector Addition Using Algebra
You still need to draw a diagram, but it can just be a quick sketch instead of a measured scale diagram.
To give the final answer, put the magnitude of the displacement and angle together:
Felicia’s displacement is 39 m [east 50° north]. Check back to the answer you got using scale diagrams and you should see it is the same.
Notebook
Redo the “self-check” example, this time using Method 2 – algebra instead of a scale diagram. Think you’ll get the same answer as we did in Method 1? Only one way to find out.
A truckload of computers travels 45 km [west] and then 65 km [south]. Find the truck’s displacement.
Suggested Answer
Step 1: Find the hypotenuse
Step 2: Find the angle
Step 3: Put the magnitude of the displacement and angle together
To give the final answer, put the magnitude of the displacement and angle together:
The truck’s displacement is 79 m [west 55° south].
Check back to the answer you got when you used scale diagrams (Method 1) and you should see it is the same.
Vector Addition (No right-angled triangle)
So far we’ve been able to add vectors that makeup a right-angled triangle. But how would you add vectors when they don’t make a right-angle triangle?
In the next example, you will learn to add vectors when you don’t have a right angle.
Example
A Toronto streetcar is traveling across the city. Find the displacement of a streetcar that travels 80.0 km [N] and then 1.0 x 10² km [W20°N].
Let’s use all three of the different methods we’ve learned to solve this type of problem:
Method 1: Using a Scale Diagram
To solve this problem, you need to choose a good scale.
In this case, a good scale is 1 cm = 10 km; therefore, 80 km = 8 cm and 100 km = 10 cm
Step 1: is complete.
Step 2: Draw the two displacement vectors, connecting them tip to tail.
Step 3: Draw another set of coordinates at the head of the first vector.
Step 4: Use the second set of coordinates (dotted lines) to draw your second vector.
Step 5: Use the second set of coordinates (dotted lines) to draw your second vector.
Method 1: Step 7
Use the scale to change the measured length of the resultant into a vector and give the final answer along with the direction.
14.7 cm x 10 km = 147 km
Use a protractor for the angle (direction).
Therefore, the displacement of the streetcar is 147 km [N41°W].
Method 2: Using Trigonometry (Sine and Cosine law)
Algebraically, when adding vectors like the above example, you can use the sine and cosine law to solve for the resultant vector. Here is the same example as before:
Find the displacement of a streetcar that travels 80.0 km [N] and then 1.0 x 10² km [W20°N].
This is how you would solve it using pen and paper.
Cosine Law: c² = a² + b² – 2ab cos C where C is the angle across from side c” and “Sine Law: a/sin A = b/sin B = c/sin C
Find magnitude
Therefore, the displacement of the streetcar is 148 km [N 39° W].
The use of this solution, when compared to the scale diagram method, will provide a more accurate and precise resultant vector.
This is why the scale diagram value is slightly different from the answer obtained using this method.
Method 3: Using Perpendicular Components
The third method of adding two or more vectors is based on resolving or breaking up a vector into perpendicular components.
This is a very common procedure – let’s review this method closely to make sure you understand how it works.
Any vector may be represented in two dimensions as having parts of the vector that lie along the x-axis and the y-axis.
For example, a displacement vector of 5.0 m [N45°E] will have a component that is parallel to the x-axis and a component that is parallel to the y-axis. Those components are perpendicular to each other.
To resolve the vector into the perpendicular components, you will use trigonometric ratios:
Now to add vectors in two-dimensions, you will resolve (or break up) all the vectors into their perpendicular components, and then add all the parallel components.
Once this is done, the Pythagorean theorem and the tangent ratio may be used to determine the magnitude of the resultant vector and its direction (angle).
Here is the same example as before but the component method will be used:
Find the displacement of a streetcar that travels 80.0 km [N] and then 1.0 x 10² km [W20°N].
[North] and [East] are the positive directions.
Therefore, the displacement of the streetcar is 148 km [N39°W].
Notice how the answer is THE SAME as the previous method’s answer.
If you are adding more than 2 vectors together, follow the same steps but once you have drawn the second vector, draw a new set of coordinates at the head of the second vector and draw the third vector from there.
Remember the resultant is still drawn from the start of the first vector to the end of the last vector.
This video shows how to draw the addition of 3 vectors.
Vector Addition Learning activity 1 of 2: Head to Tail Addition Method
Practice
Practice makes permanent. Let’s make sure you are comfortable using all of the equations taught in this learning activity. For this practice section you will need your Notebook, a ruler and protractor.
Use the chart above to decide whether to use algebra or a scale diagram.
1. A person drives 8.0 km [N], 6.0 km [W]. Find the total displacement using the Pythagorean theorem and trigonometry.
Suggested Answer
Solution: Using the Pythagorean theorem.
2. A person walks 2.0 m [E 20° S], then 4.0 m [S]. Find the total displacement using trigonometry (sine and cosine laws).
Suggested Answer
Solution: Using sine and cosine laws.
3. A truck drives 1.0 x 102 km [S], turns and drives 80.0 km [W 30° S], then turns again and drives 20.0 km [N]. Find the total displacement using the perpendicular components method.
Suggested Answer
Solution: Using perpendicular components method.
[North] and [East] directions are positive.
Self-Check
It’s that time again – get out your Notebook and try solving the following questions yourself. Then compare your answer to the suggested answers below.
- Resolve the following vectors into their components.
a.17 m/s [N]
b.40 m/s [S45°E]
a.17 m/s [N]
Draw a vector diagram that includes a directional label to represent the x and y components.
Since vector falls along the north-south vertical line, there is no horizontal component. The vector only has a vertical component.
b.40 m/s [S45°E]
Draw a vector diagram that includes a directional label to represent the x and y components.
Use the correct trigonometric formula to resolve each component
practice
Notebook
Resolve the following vectors into their components.
1. Resolve this vector into its components.
25 m/s [E]
Suggested Answer
Since there is no vertical component this vector remains as is.
2. Resolve this vector into its components.
95 m/s [N 20° W]
Vector Addition Using Components
See Vector Addition Using Components to learn about adding vectors using components.
Suggested Answer
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Self-Check: Vector Addition
Notebook
You can only add two vectors if their units and directions match. What must you do if the directions are:
a) Collinear (i.e. sharing the same line), but opposite in direction?
Suggested Answer
If two vectors are collinear, but opposite in direction, you must make their directions the same. This is done converting one of the vectors to the negative magnitude, (scalar value) with same direction as the other vector.
For example:
If vector # 1 had a direction of [E] and vector #2 had a direction of [W], then direction of vector #1 could converted from [E] to –[W]
(The negative value of direction is equal to the positive value of a direction 180° from its original heading.)
b) Non-collinear (i.e. do not have their directions along the same line)?
Suggested Answer
If the two vectors are non-collinear, you must add them, head-to-tail, and draw the resultant vector (starting from the tail of the first vector and ending at the head of the last vector). Then, you must decide how to solve the problem mathematically.
If the diagram results in a right-angled triangle, you can use the Pythagorean theorem and trigonometry functions to find the magnitude and direction of the vector.
If the triangle is not a right-angled triangle, you can
- break each vector into its x- and y-components.
- add the x-components together.
- add the y-components together.
- add the resultant x- and y-components to get a right-angled triangle that shows the resultant vector.
- Then use the Pythagorean theorem to find the magnitude of the resultant vector.
- To find the angle, you can use the inverse tangent function.
OR
You can create the triangle formed by the vectors and use a combination of cosine and sine laws. You may need to use some geometry to bring the angle back into a standard reference frame (N,S,E,W).