Vectors (ME-V1)

5 dot points across 5 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

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

A focused answer to the HSC Maths Extension 1 dot point on 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 answer →

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

A focused answer to the HSC Maths Extension 1 dot point on 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 answer →

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

A focused answer to the HSC Maths Extension 1 dot point on the 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 answer →

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

A focused answer to the HSC Maths Extension 1 dot point on 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 answer →

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

A focused answer to the HSC Maths Extension 1 dot point on 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 answer →