Further Maths - Vector Equation of a Plane

Created: January 14, 2021

2021-01-14

What is the general vector equation of a plane?
\[\pmb{r} = \pmb{a} = \lambda\pmb{b} + \mu\pmb{c}\]
\[\pmb{r} = \pmb{a} = \lambda\pmb{b} + \mu\pmb{c}\]

What must be true about the two directional vectors $\pmb{b}$ and $\pmb{c}$?

They are not parallel to one another.

PHOTO PLANE EQUATION VISUAL What equation does this photo represent?
\[\pmb{r} = \pmb{a} = \lambda\pmb{b} + \mu\pmb{c}\]
\[\left(\begin{matrix} 3+2\lambda+\mu \\ 4+\lambda-\mu \\ -2+\lambda+2\mu \end{matrix}\right) = \left(\begin{matrix} 2 \\ 2 \\ -1 \end{matrix}\right)\]

How could you rewrite this?
\[\left(\begin{matrix} 2\lambda+\mu \\\\ \lambda-\mu \\\\ \lambda+2\mu \end{matrix}\right) = \left(\begin{matrix} -1 \\\\ -2 \\\\ 1 \end{matrix}\right)\]
\[\left(\begin{matrix} 2\lambda+\mu \\\\ \lambda-\mu \\\\ \lambda+2\mu \end{matrix}\right) = \left(\begin{matrix} -1 \\\\ -2 \\\\ 1 \end{matrix}\right)\]

Why do you only need to solve two equations rather than all three?

There are only two unknowns.

\[\left(\begin{matrix} 2\lambda+\mu \\\\ \lambda-\mu \\\\ \lambda+2\mu \end{matrix}\right) = \left(\begin{matrix} -1 \\\\ -2 \\\\ 1 \end{matrix}\right)\]

If this system of equations has a solution, what does it mean?

A point lies on the plane.

If there are three points $A, B, C$ on a plane, what vectors could you also say are on the plane?
  • $\overrightarrow{AB}$
  • $\overrightarrow{AC}$
  • $\overrightarrow{BC}$

If there are three points $A, B, C$ on a plane, how could you write the plane equation?
\[\pmb{r} = A + \lambda\overrightarrow{AB} + \mu\overrightarrow{AC}\]

What is the general form of the Cartesian equation of a plane?
\[ax + by + cz = d\]

What’s the intuition for $ax + by + cz = d$?

It tests points; given an $(x, y, z)$ you can check if it’s on the plane.

What’s a normal vector to a plane?

The vector perpindicular to the plane.

\[2x + 3y + 5z = 5\]

What is the normal vector to the plane?
\[\left(\begin{matrix} 2 \\\\ 3 \\\\ 5 \end{matrix}\right)\]
\[\left(\begin{matrix} n _ 1 \\ n _ 2 \\ n _ 3 \end{matrix}\right)\]

What’s the Cartesian equation of the plane if $n$ is the normal vector?
\[n_1x + n_2y + n_3z\]

Given the start of the Cartesian equation for a plane $ax + by + cz$ and a point on the plane, how can you work out the Cartesian equation of the plane?

Substitute the point into the equation and set it equal to the result.

2021-01-18

What does it mean for points to be coplanar?

All the points lie on the same plane.

How could you prove that points are coplanar?

Come up with a plane equation using 3 of the points and use it to test the other ones.

2021-01-20

What does the Cartesian equation of a plane look like?
\[ax + by + cz = d\]

What does the parametric equation of a plane look like?
\[\pmb{r} = \pmb{a} + \lambda\pmb{b} + \mu\pmb{c}\]

What does the scalar product equation of a line look like?
\[\pmb{r}\cdot\pmb{n} = \pmb{a}\cdot\pmb{n}\]

What are the three types of plane equation?
  • Cartesian
  • Parametric
  • Scalar product

PHOTO SCALAR PRODUCT PLANE FORM What does $\pmb{n}$ represent here?

The normal vector to the plane.

Is the normal vector a plane a position of a direction vector?

A direction vector.

PHOTO SCALAR PRODUCT PLANE FORM What does $R$ represent here?

The general position vector of a point on the plane.

PHOTO SCALAR PRODUCT PLANE FORM What does $A$ represent here?

A fixed, known point on the plane.

PHOTO SCALAR PRODUCT PLANE FORM What’s the formula for $\overrightarrow{AR}$?
\[\pmb{r} - \pmb{a}\]

PHOTO SCALAR PRODUCT PLANE FORM What’s true about the line $\pmb{r} - \pmb{a}$ in relation to the normal vector $\pmb{n}$?

It is perpindicular.

PHOTO SCALAR PRODUCT PLANE FORM How would you write $\pmb{r} - \pmb{a}$ being perpindicular to the normal vector $\pmb{n}$?
\[\pmb{n}(\pmb{r} - \pmb{a}) = 0\]

Expand

\[\pmb{n}(\pmb{r} - \pmb{a}) = 0\]

?
\[\pmb{r}\cdot\pmb{n} - \pmb{a}\cdot\pmb{n} = 0\]
\[\pmb{r}\cdot\pmb{n} - \pmb{a}\pmb{n} = 0\]

How could you rewrite this?
\[\pmb{r}\cdot\pmb{n} = \pmb{a}\cdot\pmb{n}\]
\[\pmb{r}\cdot\pmb{n} = \pmb{a}\cdot\pmb{n}\]

How does $\pmb{a}\cdot\pmb{n}$ relate to the Cartesian equation of the plane?

It’s what the Cartesian equation is equal to.

\[\pmb{r}\cdot\pmb{n} = d\]

How could you rewrite this to show that the normal vector contains the coefficients of the Cartesian equation of the plane?
\[\left(\begin{matrix} x \\\\ y \\\\ z \end{matrix}\right) \cdot \pmb{n} = d\]

PHOTO PLANE PLANE INTERSECTION If the angle between the normals of two intersecting lines is $\theta$, what is the angle between the two intersecting planes?
\[180 - \theta\]

PHOTO PLANE PLANE INTERSECTION If the two normals are $\pmb{n _ 1}$ and $\pmb{n _ 2}$, what’s the formula for $\cos\theta$?
\[\cos\theta = \frac{\pmb{n_1} \cdot \pmb{n_2}{|\pmb{n_1}||\pmb{n_2}|}\]
\[\pmb{r}\cdot\pmb{n _ 1} = k _ 1 \\ \pmb{r}\cdot\pmb{n _ 2} = k _ 2\]

What is the formula for $cos\theta$, the angle between the two intersecting planes?
\[\cos\theta = \frac{\pmb{n_1} \cdot \pmb{n_2}}{|\pmb{n_1}||\pmb{n_2}|}\]

PHOTO PLANE LINE INTERSECTION If the angle between the line and the normal to the plane is $\theta$, what is the angle between the plane and the line?
\[90 - \theta\]
\[\pmb{r}\cdot\pmb{n} = k \\ \pmb{r} = \pmb{a} + \lambda\pmb{b}\]

What is the formula for $sin\theta$, the angle between the intersecting plane and line?
\[\sin\theta = \frac{\pmb{b} \cdot \pmb{n}}{|\pmb{b}||\pmb{n}|}\]
\[\pmb{r}\cdot\pmb{n} = k \\ \pmb{r} = \pmb{a} + \lambda\pmb{b}\]

What is the formula for $sin\theta$, the angle between the intersecting NORMAL TO THE plane and line?
\[\cos\theta = \frac{\pmb{b} \cdot \pmb{n}}{|\pmb{b}||\pmb{n}|}\]

PHOTO PLANE LINE INTERSECTION Why do you use $\sin$ rather than $\cos$ to tell you the angle between the intersecting plane and line?

Because $\cos\theta$ is the angle between the line and the normal, so $\sin\theta$ is $90 - \theta$.

2021-01-22

What is true about the plane equations for parallel planes?

Their normal vectors are the same.

2021-05-17

Given a point $(\alpha, \beta, \gamma)$ and a plane $ax + by + cz = d$, what’s the formula for the shortest distance from the point to the plane?
\[\frac{|\alpha a + \beta b + \gamma c - d|}{\sqrt{a^2 + b^2 + c^2}}\]

When a plane is defined as $r\cdot\pmb{\hat{n}} = k$, what does $k$ represent?

The length of the perpindicular from the origin to the plane.

What’s the general technique for finding a point $P$ reflected across a plane $\Pi$?

$P$ must lie on a line perpindicular to plane at some point $M$. You can then travel backwards the same amount to get to the other side.



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