Introduction
Introduction
A vector is a mathematical tool used to describe movement around a coordinate grid. At GCSE it is used to describe translations, and movement around various shapes.
In 2-D a vector has two parts, x and y, (x = distance moved parallel to the x-axis, y = distance moved parallel to the y-axis).
This vector can be written in two ways:
Note: Vectors are usually written in bold, but as that's a bit tricky to demonstrate in an examination. An alternative method is to underline the vector, which is how I will now write the vectors in our examples.
In 3-D the extra dimension is called the z-axis, and hence a 3-D vector is written as:
The magnitude of a vector
The magnitude of a vector is its length, and can be found using Pythagoras' Theorem.
Example:
If
then a diagram of the vector would look like:
Now you can see that the magnitude of a can be calculated using Pythagoras' Theorem:
In 3-D the result is the same.
Example:
The Direction of a vector
The Direction of a vector, θ, is the angle it makes with the positive x-axis (when measuring anti-clockwise). This is only calculated in two dimensions.
Example:
Taking the 2D example used above,
The angle to be calculated is shown below:
The angle, can be calculated using the 'tan function', therefore
The Magnitude and Direction enable us to write position vectors in cartesian form.
If we know that the magnitude of vector OA is 5 and the direction is 323.1o then the coordinates of A are:
A = (5cos323.1, 5sin323.1) = (4, -3) as expected.
In general, if the magnitude of a vector is r and the direction is then the coordinates of the point are (r cos θ, r sin θ).
Vectors can be added to represent a series of stages in a journey. The vector that represents the whole journey is called the resultant vector and is found by simply adding the x-components, then the y-components and then the z-components as follows.
This means that
And
Parallel lines run in the same direction, so parallel vectors are simply multiples of each other.
Example:
are parallel, as they can all be written as a multiple of
Notice that the last of these vectors goes in the opposite direction, but is nevertheless parallel to the other two.
In order to find the angle between two vectors we use a method called the scalar product. The scalar product (or dot product) is a bit like the 'multiple' of two vectors and works as follows.
Just multiply the x-values, then the y-values, then the z-values, and add them together.
This resulting value can then be used to find the angle between the two vectors using the formula:
where | a | = magnitude of a, | b | = magnitude of b, and θ = angle between the vectors
This rearranges to give:
Example:
The angle between the vectors
Therefore: θ = 46.0o (to 1dp)
One of the main benefits of the scalar product is that it helps us identify perpendicular vectors, which is essential for working with planes.
If two vectors are perpendicular then the angle between them is 90o.
As cos 90o = 0, the scalar product will be zero if the vectors are perpendicular.
Example:
If a and b are perpendicular, find z.
a.b = 4 + 6 +5z = 0
Therefore: 5z + 10 = 0
Therefore: z = -2
This concept is very important so make sure you understand it fully.