On this page:

- Introduction to Vectors and Scalars
- Addition and Subtraction of Vectors
- The Length and Unit Vector of a Vector
- Scalar Multiplication
- Vector Equation of a Line
- Intersections of Vectors with the X-Y Plane
- Scalar Product
- Angles between Vectors
- Vector Equations of Planes
- Angles between Planes
- Vector Product
- Area of a Vector Triangle
- Equation of a Plane given Three Points
- Minimum Distance between Two Skew Lines

**DEFINITION:** Vectors were developed to provide a
compact way of dealing with multidimensional situations without writing
every bit of information. Vectors are quantities that have magnitude and
direction, they can be denoted in three ways: in bold (**r**),
underlined (r) or
.

The position point of a vector is defined using Cartesian co-ordinates: it uses the coordinates of the OX, OY and OZ axes where O is the origin. We will be looking at vectors in 3 dimensional space in Cartesian coordinates. Similar ideas hold for vectors in n dimensional space (n-vectors).

In the above diagram r is the position vector of a point A relative to the origin O. So .

Try plotting your own vectors.

**DEFINITION:** We will look at two laws involving
addition and subtraction; The commutative law and the Associative law.

Addition and subtraction of vectors obeys the commutative law,

This means that:

In terms of the following components

For addition:

For subtraction:

This uses different routes to get to the same final desination

In terms of the following components:

**DEFINITION:** For any vector like this one

The length of a (which is denoted by |a|) is given by:

**DEFINITION:** A unit vector is a vector with unit
length 1. By standard convention we let i,
j and k be unit
vectors along the positive x, y and z axes, so in terms of components:

Find the length of your own vector and the related unit vector.

**DEFINITION:** This involves the multiplying of a
vector by a scalar, i.e. a number.

In terms of the following components:

Scalar multiplication will change the length of the vector and if the factor λ is negative the vector will point to the opposite direction. Scalar multiplication satisfies the following properties where a and b are vectors, λ and μ and are scalars.

Now try multiplying your own vectors and scalars.

**DEFINITION:** As taught in A-level and before the
equation of a straight line is given by *y = mx + c* where *m*
is the gradient and *c* is the y-intercept of the line. This is for
two dimensions. For three dimensions similar concept can be used to
represent a line in vector form.

In the diagram above, the vector (1,m) is parallel to the line AB and point A with vector coordinates (0,c) lies on the line AB. Let B be a typical point on the line with positive vector r. As d=(0,c) is a point on the line and n=(1,m) is a vector parallel to the line, the vector equation of the line AB is given by

, .

Find the vector equation of your own line by entering two points.

Continuing from above we will now look a case where a given line intersects the X-Y plane. The vector equation of the line is

If we take:

we get that

We know that in the x-y plane z = 0 so;

Substituting this into equations will give that λ = -2, which when substituted into the second and first equation give that y = -2 and x = -2. The intersection of the above line occurs at the point(-2,-2,0).

Look at another example.

Let us take two vectors.

Then the scalar product of a and b, denoted by a.b is

Setting these equal to each other gives

Using the two above formulae and setting them equal to each other as shown below we are able to calculate the angle between two vectors.

Try finding the angle between vectors yourself using the scalar product.

**DEFINITION:** Let a vector normal (vector
perpendicular to the plane) be denoted as n,
a position vector of some point in plane denoted as
a and a typical point on the plane denoted
as r.

Looking the above figure, r − a is perpendicular to n. So the formula of the vector equation of the plane is given by:

Try finding the vector equation of your own planes.

**DEFINITION:** The angle between two planes is the
angle between the two normals. The plane must firstly been written in
vector form.

For example let our plane be

In vector form this is

where

this can also be written as

If we had two planes then we would have two normal vectors say
n_{1} and
n_{2}. to find the angle between
these two vectors using the same formula when we found the angle between
vectors (above).

Find the angle between two of your own planes.

**DEFINITION:** The vector product is fundamentally different from
the scalar product. The vector product of two vectors is a vector but
the scalar product is a scalar. The vector product is given by:

where

|a| is the length of
a

θ is the angle between vectors

n is the unit vector perpendicular to
a and b whose
direction is determined by the left hand skew rule.

For vectors a × b is found by using the following:

For simplicity this can be written in terms of determinants.

Now try the vector product yourself.

**DEFINITION:**In terms of vectors the area of the
triangle below is:

Try finding the area of your own vector triangles.

Recall that the vector equation of a plane is (r - a). n = 0 where a is a point on the plane and n is a vector normal to the plane.

Suppose we have the points

which are in Cartesian form. We firstly need to find the vector parallel to the plane.

To get a vector n, which is normal to the plane, we take the vector product of the above vectors.

This gives a vector denoted by n. So the equation of the plane is found using the same method as above.

By substituting in we get

Find the equation of your plane given three of your own points.

**DEFINITION:** Skew lines are lines or vectors which
are not parallel and do not meet. We now seek the minimum distance
between these lines. By drawing a line between both lines, called a
transversal, it will be perpendicular to both lines.

The transversal connects A and B, n_{3} is the unit vector in
the direction AB and p is the required distance. As mentioned above
n_{3} is perpendicular to both n_{1} and n_{2}.

Find the minimum distance between your own lines.