Station 01 · Vector equations of lines
Lines in two dimensions
A line is a starting point plus any multiple of a direction: r = a + λb. Drag the point A and the tip of the direction vector, then slide λ.
- direction vector
- r = a + λb
- gradient and direction
- y = mx + c ⇄ vector form
- r = p + λ(q − p)
- is a point on the line?
From y = mx + c to vector form, and back
Is the point on the line?
Tests the point against the line drawn on the left.
Where is λ? The line through P and Q
r = p + λ(q − p) = λq + (1 − λ)p. Drag P and Q, then slide λ.
Station 02 · Lines that meet, lines that don't
Intersections and collinearity
Set the two vector equations equal and solve for λ and μ. Parallel lines have direction vectors that are multiples of each other.
- parallel lines
- point of intersection in 2D
- collinear points
Drag A, B (points on each line) and the arrow tips (directions).
Solving
Are P, Q, R collinear?
Drag the three points. With snapping to halves, try to line them up.
Collinearity test
Station 03 · Lines in three dimensions
Intersecting, parallel or skew?
In 3D, two lines can miss each other without being parallel. These are skew lines. Rotate the view to see it.
- r = a + λb in 3D
- parallel lines
- intersection: find λ, μ and the point
- skew lines
- points on lines, collinearity
The two lines
Points and collinearity in 3D
Station 04 · Vector equations of curves
Curves as vector equations
As the parameter t changes, the tip of r(t) traces a curve. Eliminate t to find the Cartesian equation.
- parametric curves in 2D and 3D
- identify curves from their equations
- equation from a graph
- Cartesian equations
Equation
Name that curve
Which vector equation gives this graph?
Station 05 · Circles and spheres
|r − c| = r
Every point at distance r from the centre C. In 2D that is a circle; in 3D, a sphere.
- r = r cos θ i + r sin θ j
- |r − c| = r
- circle with centre C
- sphere in 3D
- solving problems with circles and spheres
A circle in 2D
A sphere and a line in 3D
Where does the line meet the sphere?
Station 06 · The scalar product
Dot product and Cauchy–Schwarz
a · b = |a||b| cos θ measures how much two vectors point the same way. Drag the tips.
- a · b = b · a
- a · (b + c) = a · b + a · c
- k(a · b) = (ka) · b
- i · i = 1, i · j = 0
- |a · b| ≤ |a||b|
- projections
Computing a · b
Properties, checked live
The unit vectors
Station 07 · Vectors and geometry
Triangle geometry and vector proofs
The four special lines of a triangle each have a clean vector description. Then use vectors to prove geometric results.
- medians
- altitudes
- perpendicular bisectors
- angle bisectors
- geometric proofs in 2D and 3D
In vector form
Station 08 · Practice
Practice and HSC-style questions
Multiple choice, short answers and multi-part questions with a marking guide.