Vectors | HSC Extension 2

Assessment evidence

What these examples cover

These examples show assessable skills, not an official HSC topic weighting or prediction. The 2024 NSW syllabuses have their first HSC examination in 2027; students sitting the 2026 HSC continue under the applicable 2017 syllabus.

Teacher check

Teacher verification note: check the mathematics, notation, permitted technology and syllabus fit for your cohort before classroom or assessment use.

3 deterministic examples

Vectors questions with worked solutions

Each question is generated from an existing curriq QDSL archetype at a fixed seed. Marks and catalog content mappings travel with the generated question.

Worked example 1

2 marksext2-vectors-1

Area from two vectors Let

u=2vec(1,2,3)

and

v=2vec(2,-1,1)

. Find the area of the triangle formed by u, v and

u-v

Part a · 2 marks

Find the exact area.

Worked solution

Answer: ((5 sqrt(3)) 4) / 2

Vector Magnitudes And Dot Product: $abs(u)^2=14 2^2$ , $abs(v)^2=6 2^2$ and $(u dot v)^2=9 2^4$ .
Triangle From Parallelogram: ((5 sqrt(3)) 4) / 2

Catalog reference: ext2_jr25_triangle_area_from_vectors · seed 41001

Worked example 2

2 marksext2-vectors-1

Angle between vectors Find the acute angle between

bold(u)=vec(3,1,-2)

and

bold(v)=2vec(1,0,-3)

, to the nearest degree.

Part a · 2 marks

Calculate the angle.

Worked solution

Answer: (180 acos(((9 sqrt(140)) / 140))) / pi

Dot Product: $cos(theta)=9/sqrt(140)$ .
Inverse Cosine: $theta approx 40 degrees$ .

Catalog reference: ext2_morris_2025_q12a_vector_angle · seed 41001

Worked example 3

2 marksext2-vectors-1

A perpendicular vector combination Vectors satisfy

bold(a) dot bold(c)=-12

and

bold(b) dot bold(c)=3

. Find k if

bold(a)+k bold(b)

is perpendicular to

bold(c)

Part a · 2 marks

Find k.

Worked solution

Answer: 4

Dot Product Zero: $(bold(a)+k bold(b)) dot bold(c)=-4{{s}}+{{s}}k=0$ .
Solve Linear Equation: 4

Catalog reference: ext2_cssa25_perpendicular_vector_combination · seed 41001

Content mapping and common errors

Vectors content identifiers

MEX-V1

Vectors in Three Dimensions

Represent and manipulate vectors in 3D; compute dot and cross products; apply to lines and planes.

Common student mistakes

·Incorrect cross product computation (sign of the middle term)
·Confusing parallel and perpendicular conditions (dot product = 0 vs cross product = 0)
·Errors when finding the angle between lines using direction vectors

Source and methodology

Examples are deterministic curriq catalog generations from existing QDSL archetypes at the published fixed seed. The generator copies only student-facing prompts, mark allocations, topic mappings, display tables or options, checked answers and marking-path results into this resource.

Review the official NSW Curriculum source ↗

From the blog

Formula sheet

Official NESA HSC Maths Reference Sheets

Reasoning

What Makes an HSC Band 6 Maths Question?

Practice

HSC Maths Past Papers by Topic

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What these examples cover

Vectors questions with worked solutions

Worked example 1

Worked example 2

Worked example 3

Vectors content identifiers

Vectors in Three Dimensions

Source and methodology

Other Extension 2 topics

From the blog

Generate a Vectors worksheet in 30 seconds