IB Maths AA HL Ahl 3 18 Hl Intersections and Angles in 3d Questions

Practise solving line-plane and plane intersections, calculating angles with direction or normal vectors, and interpreting the resulting 3D configuration.

Exam points

Solve line–plane, two-plane and three-plane systems to find intersection points or classify the geometric solution.
Calculate angles between a line and a plane or between two planes using direction vectors and normals, then interpret the result geometrically.

[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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IB Maths AA HL Ahl 3 18 Hl Intersections and Angles in 3d Questions question 1

Question (a)

Question (b)