Vectors

Overview

A column vector can be represented as , where , , and are basis vectors (vectors with a magnitude of 1 in the , , and directions respectively).

A position vector represents a point in space, with respect to the origin. The position vector for the coordinate is .

The magnitude of a column vector is , which comes from applying the Pythagorean theorem.

Vector Equation of a Line

Given a position vector of any point on the line, and the direction vector of the line, the vector equation of that line is:

Where (lambda) is just a parameter which is varied to give different points on the line, and is the position vector of a general point on the line.

Two vector equations represent the same line if:

they are parallel (the direction vectors are scalar multiplies)
and one point from one line lies upon the other line.

Therefore, there are different vector equations for the same line. For example, all of these represent the same line (with different values of parameter for any specific point.):

(the direction vector is a scalar multiple)
(the direction vector is a scalar multiple and the position vector lies upon the other two lines)

Cartesian Equation of a Line

In two dimensions, a Cartesian equation of a line can be written as:

For the vector equation of a line, the direction vector means that the line has a gradient of in the Cartesian equation.

Changing between vector and Cartesian equations

To change between a vector and Cartesian equation, the "point slope" form of the equation can be used (where is a specific point on the line):

Alternatively, using the fact that the position vector gives coordinates of a point on the line, making the subject of both equations and setting them as equal yields the Cartesian form:

Which can then be rearranged into the desired form.

General method

In general, to find the Cartesian equation given the vector equation:

write , , and in terms of .
make the subject of each equation
equate the three equations for , giving an equation of the form .

Where all of the direction vector components are not 0:

In some cases, where one (or more) of the direction vector components is 0 (e.g. ), one of the equations will not contain , so there will be a separate equation that looks like , where one of the , , or components is a constant term.

If that didn't make much sense, imagine a 2D line . If was 0, then the line would just be , a constant. That's effectively the same idea.

Intersections of Lines

In a (2D) plane, two different straight lines either intersect, or are parallel (left). However, in 3D, two lines can be not parallel and also never intersect. These are called skew lines (right).

When two lines intersect, there are values of and such that .

Finding Intersections

To find the intersection between two lines (in the vector form):

Set the two position vectors and to be equal, and form equations for each component.
Solve the simultaneous equations to find values of and , and ensure they satisfy all equations.
Substitute or back into the equation to get the intersection point.

Similarly, to show that two lines do not intersect, show that any value of and does not hold for all three equations. The same methods apply for the Cartesian form.

Dot Product

The dot (scalar) product is one way to multiply vectors. The motivation behind the dot product is to multiply two vectors to get a scalar result.

The diagram below shows two lines with angle between them. and are vectors in the directions of the two lines, pointing away from the intersection point.

The dot product is defined as:

The is there to multiply the amount of the vectors that point in the same direction, taking the component of that lies alongside .

Another way to define for and is:

Thus, to find the angle between two vectors, we can use both definitions:

Applications

For two lines with vector equations, the angle between the two lines is equal to the angle between the direction vectors. This can be found even if the two lines do not intersect.

The dot product can be used to test if two vectors are perpendicular, as the term is equal to 0 when two vectors are perpendicular. Therefore, two vectors and are perpendicular if . Similarly, two lines are perpendicular if .

Cross Product

Given two vectors, the cross product (vector product) will produce a new vector that is perpendicular to and .

The definition of the cross product for and is:

This is given in the formula booklet. An application is the cross product of the base vectors, e.g. .

Determinant definition of the cross product

Alternatively, the cross product can be worked out using the determinant of a 3x3 matrix:

Application to 3x3 matrix inverse

The cross product can be used to find the inverse of a 3x3 matrix, by finding the cross products of columns of :

Find the first row of by
Find the second row of by (note how this is the other way round)
Find the third row of by
Divide by