KS4. Vectors

Vectors is part of the extended iGCSE syllabus. It basically introduced a new part of geometry which uses a very efficient language to describe position and movement.

The essence of a vector is very similar to the essence of a coordinate. It is specified by a horizontal and a vertical component that combine to define a specific position or a specific movement. The vector can also be specified by drawing a straight line in a specific direction.

Let’s start by combining vectors using addition and scalar multiplication.

Addition works the way we would expect it to work. So if we are dealing with lines, to add two lines we replace them with a line that goes from the tail of the first line to the head of the second line. With vectors written using numbers its even easier, we simply add the numbers.

With scalar multiplication we keep the direction the same and increase the length of the line by a scale factor of the number we are multiplying by. If the vector is given as numbers, we multiply each of the numbers (the components) by the scalar multiple.

Exercise

Let’s practice this with geometrical vectors by completing exercises 5 and 6 on pages 291 to 295 of the extended textbook.

The answers are below:

Now we will try some arithmetic with the vectors. As we mentioned above, this is fairly straightforwards. A negative vector goes in exactly the opposite direction to a positive vector, and subtracting a vector is equivalent to adding its negative vector (as is the case with arithmetic).

We should know that if one vector is a scalar multiple of another vector then the two vectors are parallel or the same.

Exercise

Let’s try exercises 7 and 8 from pages 297 to 299 of the extended text book:

The answers are below:

Modulus of a vector

The modulus of a vector is the length of that vector, and we can find it using Pythagoras’ theorem. Let’s look together at a couple of examples.

Exercise

Now let’s complete exercise 9 from pages 300 to 301 of the extended textbook:

The answers are below:

Vector Geometry

The great power of vectors is that they enable us to more easily solve some quite complicated geometrical problems. Let’s see how they can be used in this way in the following example: