IB Math · Topic practice · IB Mathematics AA HL · Unit 3
Geometry & Trigonometry (AHL) A mixed exercise set, like a textbook: every problem type this unit covers, shuffled together and getting harder as you go. Work on paper, reveal to check — and hit
All levels Level 1 · Basics Level 2 · Standard Level 3 · Exam
1. Find the magnitude of the vector ( 1 2 2 ) \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix} 1 2 2 .
2. A line has vector equation r = ( − 3 ; − 2 ; − 5 ) + t ( 1 ; 2 ; 2 ) \mathbf{r} = (-3; -2; -5) + t(1; 2; 2) r = ( − 3 ; − 2 ; − 5 ) + t ( 1 ; 2 ; 2 ) . Find the point at t = 2 t = 2 t = 2 — give its x x x -coordinate.
3. The vectors ( 2 3 3 ) \begin{pmatrix} 2 \\ 3 \\ 3 \end{pmatrix} 2 3 3 and ( 3 1 k ) \begin{pmatrix} 3 \\ 1 \\ k \end{pmatrix} 3 1 k are perpendicular. Find k k k .
4. Let u = ( 3 4 ) \mathbf{u} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} u = ( 3 4 ) and v = ( 4 3 ) \mathbf{v} = \begin{pmatrix} 4 \\ 3 \end{pmatrix} v = ( 4 3 ) , and let θ \theta θ be the angle between them. Find cos θ \cos\theta cos θ .
5. Using a compound-angle identity, find the exact value of sin 15 ∘ \sin 15^\circ sin 1 5 ∘ .
6. For u = ( − 4 ; − 4 ) \mathbf{u} = (-4; -4) u = ( − 4 ; − 4 ) and v = ( 3 ; 4 ) \mathbf{v} = (3; 4) v = ( 3 ; 4 ) , find the scalar projection of u \mathbf{u} u onto v \mathbf{v} v .
7. Find the magnitude of the vector ( − 2 6 9 ) \begin{pmatrix} -2 \\ 6 \\ 9 \end{pmatrix} − 2 6 9 .
8. A line has vector equation r = ( − 2 ; − 1 ; − 4 ) + t ( 1 ; 2 ; 2 ) \mathbf{r} = (-2; -1; -4) + t(1; 2; 2) r = ( − 2 ; − 1 ; − 4 ) + t ( 1 ; 2 ; 2 ) . Find the point at t = 3 t = 3 t = 3 — give its x x x -coordinate.
9. The vectors ( 1 − 4 − 2 ) \begin{pmatrix} 1 \\ -4 \\ -2 \end{pmatrix} 1 − 4 − 2 and ( 4 − 1 k ) \begin{pmatrix} 4 \\ -1 \\ k \end{pmatrix} 4 − 1 k are perpendicular. Find k k k .
10. Let u = ( 4 3 ) \mathbf{u} = \begin{pmatrix} 4 \\ 3 \end{pmatrix} u = ( 4 3 ) and v = ( 8 15 ) \mathbf{v} = \begin{pmatrix} 8 \\ 15 \end{pmatrix} v = ( 8 15 ) , and let θ \theta θ be the angle between them. Find cos θ \cos\theta cos θ .
11. Using a compound-angle identity, find the exact value of cos 7 π 12 \cos \frac{7\pi}{12} cos 12 7 π .
12. For u = ( − 4 ; − 1 ) \mathbf{u} = (-4; -1) u = ( − 4 ; − 1 ) and v = ( 5 ; 12 ) \mathbf{v} = (5; 12) v = ( 5 ; 12 ) , find the scalar projection of u \mathbf{u} u onto v \mathbf{v} v .
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