Intersections and angles between lines and planes (HL only) — IB Mathematics AA HL
IB Mathematics AA HL · Geometry & Trigonometry · 15 min read
1. Intersection of a Line and a Plane★★★☆☆⏱ 5 min
Exam tip: Always check which case you have before concluding there is no intersection or an infinite number of intersections.
2. Angle Between a Line and a Plane★★★★☆⏱ 5 min
The angle between the line and the plane's normal is $90^\circ - \theta$, so using the dot product formula for the angle between two vectors gives the identity $\cos(90^\circ - \theta) = \sin\theta = \frac{|\vec{d} \cdot \vec{n}|}{|\vec{d}||\vec{n}|}$. The absolute value ensures we get an acute angle.
3. Angle Between Two Intersecting Planes★★★★☆⏱ 5 min
IB exam questions almost always ask for the acute angle between two planes. Only omit the absolute value if explicitly asked for the obtuse angle.
Common Pitfalls
Why: You confuse the angle between the line and the normal with the line-plane angle itself
Why: The dot product of normals can be negative, which means the angle between the normals is obtuse; the dihedral angle is always acute unless stated otherwise
Why: Rushing in the exam, you misread the question which asks for the intersection, not just the parameter
Why: You forget the three possible cases for line-plane intersection