Angles Between Lines & Planes | DP IB Maths: AA HL Revision Notes 2021

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Angle Between Line & Plane

What is meant by 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

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

You need to know:
- A direction vector for the line (b)
  - This can easily be identified if the equation of the line is in the form $r = a + λ b$
- A normal vector to the plane (n)
  - This can easily be identified if the equation of the plane is in the form $r \cdot n = a \cdot n$
Find the acute angle between the direction of the line and the normal to the plane
- Use the formula
  - The absolute value of the scalar product ensures that the angle is acute
Subtract this angle from 90° to find the acute angle between the line and the plane
- Subtract the angle from $\frac{π}{2}$ if working in radians

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

Examiner 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 I 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
$\cos θ = \frac{n_{1} \cdot n_{2}}{|n_{1}| |n_{2}|}$

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

Examiner 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

Angles Between Lines & Planes (DP IB Maths: AA HL)

Revision Note

What is meant by the angle between a line and a plane?

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

How do I find the angle between two planes?

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1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

1.2 Exponentials & Logs

1.2.1 Introduction to Logarithms

1.2.2 Laws of Logarithms

1.2.3 Solving Exponential Equations

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.3.5 Compound Interest & Depreciation

1.4 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

2.2.2 Factorising & Completing the Square

2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

2.3 Functions Toolkit

2.3.1 Language of Functions

2.3.2 Composite & Inverse Functions

2.3.3 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions

2.7.4 Roots of Polynomials

2.8 Inequalities

2.8.1 Solving Inequalities Graphically

2.8.2 Polynomial Inequalities