Back To Course

OUP Oxford IB Math Studies: Online Textbook Help11 chapters | 129 lessons

Instructor:
*Matthew Bergstresser*

There is an equation to determine the angle between a line and a plane. In this lesson, we will go through the derivation of the equation and an example using the equation.

If you have ever seen a sundial, you have seen a classic example of a line intersecting a plane. Actually, the part of the sundial that intersects the plane is a plane itself called the gnomon. It is very common for the gnomon to be a right triangle.

Let's pretend the hypotenuse of the right triangle is the line that intersects the circular plane, which is the base of the sundial.

Let's go through the derivation for the equation to determine the angle between a line intersecting a plane.

A mathematical derivation of a geometric scenario usually starts with a diagram so let's make one!

Here the plane is shaded in gray and the dashed, black line lies on the plane. The dashed, purple arrow is normal to the plane, and the solid, red arrow is the line that intersects the plane. θ is the angle we are looking to find. Φ is the angle between the line and the normal.

There are a couple of steps to determine θ, the angle between the red arrow and the plane. First, we need to determine the angle between the red arrow and the dashed, black line, which we labeled Φ.

We can say there are vectors that lie along the red line and the normal line. A **vector** has a length and a direction. Since we are switching from two lines to two vectors, we can use an equation to determine Φ, which is:

Let's see how to determine the value in the numerator.

The red vector in our diagram in the notation:

where l, m and n represent the x, y and z components of the vector.

The equation for a plane is:

We don't need the *D* term in the equation of the plane. All we need from that equation are the *A*, *B* and *C*, which are the coefficients of x, y and z. We'll use the values from the vector and the plane we just discussed to determine the dot product between the vectors. The dot product multiples all x-terms together, y-terms together, and z-terms together and then sums the results. This is what it looks like:

Next, we'll look at how to determine the denominator of our original equation for the angle between two vectors.

The denominator of our equation is the product of the magnitudes of each vector. To determine the magnitude of the vectors, we use the Pythagorean theorem and include the values of all three dimensions. The denominator of our equation is:

Combining the numerator and denominator we get the equation for the angle between the red vector and the vector normal to the plane, which is:

Hmm, we did all of that work and didn't get the angle between the red line and the plane! We got Φ, not θ! But, don't worry, we can get this equation to represent θ, with one quick step. Let's look back at our diagram.

The sum of angles θ and Φ is 90°. The sine and cosine functions are 90° out of phase, which means the cosΦ = sinθ. All we have to do is swap these terms in the equation we just derived! Our final equation for the angle between the red arrow and the plane is:

Now, let's put that equation to use and determine the angle between a line and a plane with an example.

A line passes through the points (2, -1, 3) and (0, -1, -4). Find the angle of intersection of this line with the plane x - 4y + 3z - 5 = 0.

To get the vector that lines along the line passing through the two points, all we have to do is subtract the x-terms, y-terms and z-terms.

The vector perpendicular to the plane is:

The equation for the angle between the line and the plane is:

Calculating the numerator, we get:

The denominator is:

Plugging these values into the equation, we get the angle between the line and the plane.

The angle between a line and a plane is:

The numerator is the dot product between the line, which we turn into a **vector** (which has a length and a direction), and the vector normal to the plane. The denominator is the product between the magnitudes of both vectors. The magnitude of a vector can be determined using the Pythagorean theorem.

The vector lying along the line that intersects the plane can be determined by subtracting the x-terms, y-terms and z-terms of the two points that lie along the line.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
8 in chapter 10 of the course:

Back To Course

OUP Oxford IB Math Studies: Online Textbook Help11 chapters | 129 lessons

- What Are Platonic Solids? - Definition and Types 4:39
- Prisms: Definition, Area & Volume 6:12
- Pyramids: Definition, Area & Volume 7:43
- What Are Cylinders? - Definition, Area & Volume 5:09
- Cones: Definition, Area & Volume 8:59
- Spheres: Definition, Area & Volume 5:22
- How to Find the Distance Between Points on a Solid
- Angle Measurement Between Lines & Planes
- Go to OUP Oxford IB Math Studies Chapter 10: Geometry & Trigonometry 2

- Introduction to HTML & CSS
- Introduction to JavaScript
- Computer Science 332: Cybersecurity Policies and Management
- Introduction to SQL
- Computer Science 203: Defensive Security
- JavaScript Language Basics
- Error Handling, Debugging & Events in JavaScript
- HTML Elements & Lists
- Conditionals, Arrays & Loops in JavaScript
- Introduction to HTML
- Anti-Bullying Survey Finds Teachers Lack the Support They Need
- What is the ASCP Exam?
- ASCPI vs ASCP
- MEGA Exam Registration Information
- MEGA & MoGEA Prep Product Comparison
- PERT Prep Product Comparison
- MTLE Prep Product Comparison

- Simple Scientific Tools & Uses for Kids
- Chi Square Distribution: Definition & Examples
- Stars: Definition & Facts
- Linear Approximations Using Differentials: Definition & Examples
- Access Control: Types & Implementation
- 'I Am' Poem Lesson Plan
- Key Controls in Cybersecurity Risk Management: Definition & Use
- Quiz & Worksheet - Line Integrals
- Quiz & Worksheet - Frankenstein Creature Quotes
- Quiz & Worksheet - A Christmas Carol Facts
- Quiz & Worksheet - Preschool Classroom Technology
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- ESL Conversation Questions
- 5th Grade Math Worksheets

- Praxis Biology (5235): Practice & Study Guide
- Praxis Biology and General Science: Practice and Study Guide
- UExcel Research Methods in Psychology: Study Guide & Test Prep
- UExcel Business Law: Study Guide & Test Prep
- MTTC School Counselor (051): Practice & Study Guide
- SBA Math - Grade 8: Introduction to Decimals
- Biology Review
- Quiz & Worksheet - Thyroid Disorders & Thyroidectomies
- Quiz & Worksheet - Hemorrhage
- Quiz & Worksheet - General and Special Relativity Theories
- Quiz & Worksheet - Pluto, Eris, Haumea & Ceres
- Quiz & Worksheet - The Creation of Magnetic Fields

- The Doppler Effect: Definition, Examples & Applications
- The Coordinated School Health Program: Definition & Model
- 2nd Grade Word Walls
- How to Pass a Psychology Test
- How Long Does it Take to Learn French?
- Kindergarten Word Walls: Ideas & Activities
- Illinois Common Core Standards
- Thanksgiving Bulletin Board Ideas
- Light for Kids: Activities & Experiments
- Curriculum Development Templates
- Fun & Easy Science Experiments for Kids
- Multiplication Games for Kids

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject