IB Maths AA HL 3.18 Intersections and angles in 3D Question Bank

Practise IB Mathematics HL 3.18 by applying intersections and angles in 3d methods to exam-style questions.

Exam points

identify the mathematical structure, variable or representation
select and apply the correct theorem, formula or algorithm
check the result using units, domain, graph or logical reasoning

[Maximum number: 7]

(a)

The z-axis meets the plane $\Pi$ at the point P. Find the coordinates of P.

[ 2 ]

On the z-axis, x=y=0. (M1) Thus -11z=64, so
$\boxed{P=\left(0,0,-\frac{64}{11}\right)}.\quad (A1)$

(b)

Find the angle between the line $\frac{x-2}{3}=\frac{y+5}{4}=\frac{z}{2}$ and the plane $\Pi$ .

[ 5 ]

For line direction (3,4,2) and plane normal (16,24,-11),
$122=\sqrt{29}\sqrt{953}\cos\theta.\quad (M1)(A1)(A1)$
Thus $\theta=42.8^\circ$ is the angle with the normal. (A1) The angle with the plane is therefore
$\boxed{90^\circ-42.8^\circ=47.2^\circ}\quad(0.824\text{ rad}).\quad (A1)$
Note: The equivalent sine formula is accepted.

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AHL 3.18 (HL)—Intersections and angles in 3D question 1

Question (a)

Question (b)