3.11.3 Angles Between Lines & Planes

Angle Between Line & Plane

How do I find the angle between a line and a plane?

When you find the angle between a line and a plane you will be finding the angle between the line itself and the line on the plane that creates the smallest angle with it
- This means the line on the plane directly under the line as it joins the plane
It is easiest to think of these two lines making a right-triangle with the normal vector to the plane
- The line joining the plane will be the hypotenuse
- The line on the plane will be adjacent to the angle
- The normal will the opposite the angle
As you do not know the angle of the line on the plane you can instead find the angle between the normal and the hypotenuse
- This is the angle opposite the angle you want to find
- This angle can be found because you will know the direction vector of the line joining the plane and the normal vector to the plane
- This angle is also equal to the angle made by the line at the point it joins the plane and the normal vector at this point
The smallest angle between the line and the plane will be 90° minus the angle between the normal vector and the line
- In radians this will be $\frac{π}{2}$ minus the angle between the normal vector and the line

3-11-3-angle-between-a-line-and-a-plane-diagram-1

Exam Tip

Remember that if the scalar product is negative your answer will result in an obtuse angle
- Taking the absolute value of the scalar product will ensure that you get the acute angle as your answer

Worked Example

Find the angle in radians between the line L with vector equation $r = (2 - λ) i + (λ + 1) j + (1 - 2 λ) k$ and the plane $Π$ with Cartesian equation $x - 3 y + 2 z = 5$ .

3-11-3-ib-hl-aa-angle-line-and-plane-we-solution-1

Angle Between Two Planes

How do we find the angle between two planes?

The angle between two planes is equal to the angle between their normal vectors

It can be found using the scalar product of their normal vectors

If two planes Π₁ and Π₂ with normal vectors n₁ and n₂meet at an angle then the two planes and the two normal vectors will form a quadrilateral

The angles between the planes and the normal will both be 90°
The angle between the two planes and the angle opposite it (between the two normal vectors) will add up to 180°

3-11-3-ib-hl-aa-angle-between-two-planes-diagram-2

Exam Tip

In your exam read the question carefully to see if you need to find the acute or obtuse angle
- When revising, get into the practice of double checking at the end of a question whether your angle is acute or obtuse and whether this fits the question

Worked Example

Find the acute angle between the two planes which can be defined by equations $Π_{1} : 2 x - y + 3 z = 7$ and $Π_{2} : x + 2 y - z = 20$ .

3-11-3-ib-hl-aa-angle-between-two-planes-we-solution-2

DP IB Maths: AA HL

Revision Notes

3.11.3 Angles Between Lines & Planes

Angle Between Line & Plane

How do I find the angle between a line and a plane?

Exam Tip

Worked Example

Angle Between Two Planes

How do we find the angle between two planes?

Exam Tip

Worked Example

Author: Amber

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