Vectors (ME-V1)

6 dot points across 6 inquiry questions, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How do we combine a velocity relative to the air or water with a constant wind or current, and find the resulting speed, direction or required heading?

Solve relative velocity problems involving constant crosswind/cross-current using vector diagrams, and describe the direction of a vector where required

The Year 12 Extension 1 answer on relative velocity, new in the 2024 syllabus: ground velocity equals velocity relative to the air or water plus the wind or current, how to find the resultant speed and direction with a vector diagram or components, the heading needed to travel straight across or along a set track, and how to state directions as bearings.

12 min read →

How do we add, subtract and scale two-dimensional vectors, and how do we find their magnitudes?

Perform vector arithmetic with vectors in the plane, including component and column-vector notation, and find the magnitude and unit vector

Vector arithmetic. Vectors as arrows, head-to-tail and parallelogram addition, subtraction, scalar multiplication, magnitude by Pythagoras, unit vectors, the standard notation conventions, the triangle inequality, and worked examples with stage-by-stage diagrams.

14 min read →

What is the scalar (dot) product of two vectors, and what does it measure?

Compute the scalar product of two vectors in component or geometric form and use it to find the angle between vectors and test orthogonality

Scalar product. The component formula a dot b = a1 b1 + a2 b2, the geometric formula |a||b| cos theta, why the two agree, the sign of the dot product, bilinearity, finding the angle between vectors and testing orthogonality, with worked examples and diagrams.

13 min read →

How do we find the projection of one vector onto another, and what does it mean geometrically?

Compute the scalar and vector projection of one vector onto another and interpret it geometrically

Vector projection. The scalar projection and the vector projection, their formulas, the stage-by-stage construction (drop a perpendicular, mark the right angle, draw the projection vector), the parallel and perpendicular decomposition, and applications, with worked examples.

13 min read →

How do we use vectors to prove standard geometric results in the plane?

Use vector methods to prove geometric properties, including parallelism, perpendicularity, midpoint and ratio division

Geometric proofs using vectors. Position vectors and the four-step tactic, parallelism and perpendicularity tests, the midpoint and section formulas, the parallelogram and rectangle criteria, and complete worked proofs of the midpoint connector theorem, the diagonal-bisection property and the cosine rule, with diagrams.

14 min read →

How do we represent a line in the plane using vector equations?

Write parametric vector equations of lines and convert between vector and Cartesian forms

Parametric vector equations of lines. The point-direction form r = a + t d built up stage by stage, the line through two points, conversion to and from Cartesian, vertical and horizontal lines, intersections, parallel and coincident lines, and collision versus path-crossing, with worked examples and diagrams.

14 min read →