Vectors

A vector is a quantity in mathematics and physics that has a size (called a magnitude) and a direction. A quantity with only a size or magnitude is called a scalar. Temperature is an example of a scalar, as is your age.  The distance between two objects in a straight line, if you don't care about the direction, is a scalar.  The distance between two objects following any path is the distance along that path. However, the distance required between two objects in a straight line when we do care about the direction, and include the direction in the description, is called the displacement. Displacement is a vector.

The physics uses of vectors will be described elsewhere. This page is a brief mathematical introduction to vectors.

A simple vector

Vectors are represented graphically using arrows. The length of the arrow represents the magnitude of the vector, and the direction of the arrow (as illustrated by the head of the arrow) represents the direction of the vector. Here we will demonstrate two-dimensional vectors. Three-dimensional vectors are not really a part of this course, but you can extrapolate how they work from 2D vectors.

Access (in a different browser tab, window or device) this excellent interactive simulation by Tom Walsh that uses Geogebra to illustrate physical concepts. His home page, oPhysics, is here. You should see something like this:

Click on the end of the red arrow and drag it to the origin of the graph so it looks like this:

If you now click and drag on the end of the blue arrow you can move the vector around. If you drag to x = 2, y= 4 you should see this.

Note that it takes two numbers to describe a two-dimensional (2D) vector. In the example above you can see the angle of the vector above the horizontal x-axis is 63.43 degrees and the length of the vector is 4.47 units. Drag the arrow using the blue circle to other locations to see how the angle and the magnitude changes.

Vector addition

Drag the red circle away from the origin of the graph. Adjust the red and blue circles until your chart resembles this one:

Note that we now have two vectors, the red vector and the blue vector. The black vector is the result of adding the two vectors together. The number at the top represents the combined length of both vectors and the angle is the angle of the combined, or resultant, vector.  The dotted line at the end of the red vector is the same magnitude and direction as the blue vector, and the dotted line at the end of the blue vector is the  is the same shape and direction as the red vector. Where the dotted lines meet is the end of the resultant vector. This graphical method of adding vectors is known as the Parallelogram Method. 

Adjust the red and blue vectors to see how it works. You can try selecting the Triangle method for an alternative. Learn through play.

Component vectors

With "Show Parallelogram Method" selected, move the blue vector so that it lies along the x-axis with a value of 4. Move the red vector so it lies along the y-axis with a value of 3. You should see this:

Note that the resultant vector has a magnitude of 5 and an angle of 36.87 degrees. Any possible 2D vector can be created using two vectors, one on the x-axis and one on the y-axis. Try it. These two axis-aligned vectors are called the component vectors of the resultant vector. Component vectors are easy to describe because their direction is always the same (along the x or y axes) and all that varies is their magnitude. In another different browser tab, window or device, open this simulation to play with vector components.

Calculating the component vectors

You could do this graphically. Draw the resultant vector and then project onto the x-axis and the y-axis to find the component vectors. However, it is generally easier, quicker and more accurate to use trigonometry.  Any two component vector and the result vector will form a right-angled triangle, so we can use sine, cosine and Pythagoras to work out the component vectors.  Use the Vector Components interactive to create this vector (A, the length, equals 50 and θ, the angle is 36.9):

This is a 3-4-5 triangle. The base, or x-axis component vector, is 30 units long, the height, or y-axis component vector, is 40 units long and the hypotenuse, or the length of the resultant vector, is 50 units long.  Knowing the length and angle of the result vector we can work back to find the lengths of the x-axis and y-axis component vectors using sin and cos. From the IB Diploma Physics data booklet (2023) we get:

Where AH is the x-axis component (H for horizontal) and AV is the y-axis component (V for vertical). If A = 50 units and θ = 36.9 then A cos θ = 40 and A sin θ = 30. Note that in science we sometimes use degrees for angle and sometimes (more frequently later in the course) we use radians. Make sure your calculator is set correctly accordingly.

Combining component vectors

Again trigonometry and Pythagoras are your friends. If you know the lnegth of two component vectors then the angle can be calculated using tan. From the example above, if the lengths are 30 and 40 units then the inverse tan of 30/40 = 36.87 degrees. Similarly 30² + 40² = 2500, and the square root of 2500 is 50.

Adding vectors using component vectors

Earlier we described adding vectors graphically using the Parallelogram or Triangular methods. However, at this level we expect you can add using the component vectors. Suppose you have two vectors, A and B. For both you know the magnitude and the angle. The steps are:

1. Calculate the component vectors of A.

2. Calculate the component vectors of B.

3. Add the two x-axis values (note that one or all of the component values and the resultant value might be negative).

4. Add the two y-axis values (note that one or all of the component values and the resultant value might be negative).

5. Use the new x-axis and y-axis values to calculate the magnitude and direction of the resultant vector.

Subtracting vectors is done by reversing one of the vectors and then adding them. In this case the magnitude of the second vector remains unchanged, but the angle increases by 180 degrees (or π radians).

For graphical practice on adding and subtracting vectors play this game.