Algebra

• Arithmetic & Geometric Series
• Binomial Theorem
• Completing The Square

Functions

• Composite Function
• Concept And Notation
• Exponentials & Logarithms
• Factor Theorem
• Inverse Function
• Quadratic Functions
  • Quadratic Forms
• Transformations

Trig

• Area of Sectors
• Circular (Trig) Functions
• Oblique Triangles
• Radian Measure
• Right Angle Trig
• Right Angle Trig on the Unit Circle
• Tangent and Slope of Lines
• Trig Identities

Matrices

• Determinants and Inverses
• Matrix Definitions
• Matrix Operations
• System of Equations

Vectors

• 2D and 3D Lines
• Basic Vector Arithmetic
• Intro To Vectors
• More on Lines
• Scalar Product

Stats

• Continuous Variables
• Discrete Variables
• Standard Deviation
• Statistics Representations
• StatsVocabulary
• Using Normal Distributions

Probability

• Binomial Distribution
• Probability Rules
• Representations of Probability

Calculus

• Derivatives by First Principle
• Intro to Derivatives
  • Derivatives Of More Complex Functions
  • Derivatives Part 1
  • Derivatives Part 2
• Intro To Integration
  • Indefinite Integrals
• Kinematics
• Limits
• Local Max and Min
• Optimization
• Tangent and Normals
• Volumes of Revolution

There are two ways to determine if two lines are parallel. Both involve the direction vector. The equation below provides the value of the cosine of the angle between two vectors.

If the cosine of the angle between the two direction vectors is 1 or -1 then the angle between the vectors is 0 (or 180) and the vectors are parallel (anti-parallel).

The second way can be done almost by inspection. That is if the direction vectors are scalar multiples of each other then they are parallel. For example the vectors:

The vector u is 3 times vector v.

This one is pretty easy. If the cosine of the angle between the direction vectors is 0 then the angle between them is 90 or 270 and the lines are perpendicular. Again we use the scalar product formula:

We can go one step further, since $cos(90)=0$ and say that for perpendicular lines:

Its important to note that in 3D perpendicular lines do not necessarily intersect.

A cartesian equation is one that is only in terms of x and y (and z if 3D). There is no t variable in the equation.

Lets start with a 2D example. Lets write a Cartesian equation from the following vector equation:

The first step is to write two separate equations, each from a different row in the vector equation.

Solving both for t:

Setting them equal with a little rearrangement we have:

This is the "Cartesian Equation." Someone please correct me if I'm wrong SL folks do not need to write a Cartesian Equation for a 3D line.

Case 1 - Finding the intersection point is straight forward if you have Cartesian equations for both lines. If you do not have Cartesian equations for both lines it might be worth finding them…

With two Cartesian equations you simply find the intersection of the two lines as you would in a basic Algebra course. Use your GDC or solve one equation for y (or x) and substitute into the second equation. Voila!

Case 2 - 3D vectors… Each line will have its own "time" variable. Often represented by t and s when there are two vector equations. If the two lines intersect then there is a value for t and a value for s that will results in the same (x,y,z) coordinates for both lines. Essentially we set the two equations equal to each other and solve for t and s.

Example of Find an Intersection Point

Given the two vector equations $\binom{x}{y}=\binom{1}{3}+t \binom{3}{-1}$ and $\binom{x}{y}=\binom{2}{2}+t \binom{-1}{3}$ find the point $(x,y)$ where they intersect.

First we find the Cartesian equations. We found the first above to be:

Similarly we can find a Cartesian equation for the second:

Solving for t and setting the two equation equal:

Simplifying:

Solving the second Cartesian equation for y ($y=8-3x$) and substituting into the first equation we get:

Resulting in $x=\frac{14}{8}=\frac{7}{4}=1.75$ and $y= 2.75$. We can confirm this graphically as shown below.

Example of Finding Intersections in 3D

Given the two vector equations

and

Since we only have two unknowns to solve for t and s we only need two equations. Lets take the first two rows of each equation:

Each of the x-equations can be set equal and each of the y-equations set equal. This results in two equations and two unknowns. Solve… We find that $t=-3, s=-2$. One of these values is then substituted back into the corresponding original equation to solve for the (x,y,z) coordinates. Substituting $t=-3$ into the first equation:

You can also check your work by substituting $s=-2$ into the other equation and if all is well you will get the same coordinate point for the intersection.

Two objects will collide ONLY IF they have the same coordinates (x,y) at the same time t. Think about it… Two cars can have the same position, its only a problem if they do so at the same time.

If there is an intersection point then a collision is possible. You must check that the intersection points occurs when the two time variables are the same. In the example above there would be no collision as the time variables (s and t) are not the same.