MCV4U · Calculus and Vectors · Sunway CIMP

Lines and Planes

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Topic H Course Culminating Task Taught by Yuxiang (Kurath) Ji & Lucas
A line crossing a plane in three-space Coordinate axes with a shaded plane cutting across them. A straight line passes through the plane at a single marked point, drawn faint and dashed where it lies behind the plane, and the plane's normal vector stands perpendicular at the crossing. A dashed drop line runs from the crossing point to its shadow on the xy-plane. x y z π n L

Most of the topic sits in this one picture. A line meets a plane at a single point when \(\vv{m} \cdot \vv{n} \neq 0\). When that dot product is zero the line is parallel to the plane, and then everything depends on whether one point of the line satisfies the plane's equation.

Contents

The five lessons

01Equations of Lines in R² and R³ 02Equations of Planes 03Plotting and Sketching in R³ 04Intersections and Systems 05Distances ·Two Recorded Lectures

At a glance

The forms you will keep coming back to

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Practise

Then test yourself

Eight problems Homework Every subtopic on this site gets a dedicated problem, and every solution is revealed one step at a time so you can stop as soon as you are unstuck. Open homework Four quizzes Assessments Self-marking, with instant feedback and a score broken down by achievement category. Communication questions stay open-response, with a model answer and a marking rubric. Open assessments 70 marks Unit Test A full unit test across Knowledge, Application, Thinking and Communication, with mark allocations, a suggested time, and complete typed solutions. Open the unit test

How the assessments are built

The four achievement categories

Knowledge

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Application

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Thinking

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Communication

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Every one of the four assessments carries questions from all four categories, and so does the unit test.

Go further

External resources

Paul’s Online Math Notes A clean, exam-style treatment of equations of lines and planes, with many more worked examples in the same notation used here. Read
GeoGebra 3D Calculator Type a plane or a line and rotate it. The fastest way to check that a sketch you have drawn by hand looks like the object. Visualise
Khan Academy: Vectors and Spaces Video explanations of the underlying vector algebra: dot products, cross products, projections and linear combinations. Watch