Planes in Space

Name: Planes in Space
Author: StemPy Lab

A flat roof, a wall and a sheet of glass are all pieces of planes. A plane is fixed by one point and one direction that stands straight out of it. Write the equation of a plane in vector and Cartesian form. Find the angle between two planes and between a line and a plane. Measure the distance of a point from a plane, and find where two planes meet. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level Further Maths and IB.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
Equations of a plane
A plane is fixed by one point and a normal vector, which stands at right angles to the plane. If $\vec{n}$ is a normal and $\vec{a}$ is a point of the plane, then $\vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n}$ . In coordinates this is $Ax + By + Cz + D = 0$ , and $(A, B, C)$ is a normal.
1 simulation1 practice question1 Python program3 problems1 Python lab
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Lesson 2about 25 min
Angles: plane with plane, and line with plane
The angle between two planes is the angle between their normals, taken acute: $\cos\theta = \dfrac{\lvert \vec{n}_1 \cdot \vec{n}_2 \rvert}{\lvert \vec{n}_1 \rvert \lvert \vec{n}_2 \rvert}$ . The angle $\varphi$ between a line with direction $\vec{b}$ and a plane with normal $\vec{n}$ has $\sin\varphi = \dfrac{\lvert \vec{b} \cdot \vec{n} \rvert}{\lvert \vec{b} \rvert \lvert \vec{n} \rvert}$ .
1 simulation1 practice question1 Python program3 problems
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Lesson 3about 25 min
Distance of a point from a plane and the line of intersection of two planes
The distance of the point $(x_1, y_1, z_1)$ from the plane $Ax + By + Cz + D = 0$ is $\dfrac{\lvert Ax_1 + By_1 + Cz_1 + D \rvert}{\sqrt{A^2 + B^2 + C^2}}$ . Two planes that are not parallel meet in a line. Its direction is $\vec{n}_1 \times \vec{n}_2$ , and a point of it is found by putting one coordinate equal to $0$ .
1 simulation1 practice question1 Python program4 problems1 Python lab1 question set1 note
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Planes in Space: Chapter Test

20 questions · 50 min

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Planes in Space

Course outline (3 lessons)

Equations of a plane

Angles: plane with plane, and line with plane

Distance of a point from a plane and the line of intersection of two planes

Test Your Learning

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