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.
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
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.
What does $R$ represent here??
The general position vector of a point on the plane.
What does $A$ represent here??
A fixed, known point on the plane.
What’s the formula for $\overrightarrow{AR}$??
$$ \pmb{r} - \pmb{a} $$
What’s true about the line $\pmb{r} - \pmb{a}$ in relation to the normal vector $\pmb{n}$??
It is perpindicular.
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??
$$ \left(\begin{matrix} x \\ y \\ z \end{matrix}\right) \cdot \pmb{n} = d $$
If the angle between the normals of two intersecting lines is $\theta$, what is the angle between the two intersecting planes??
$$ 180 - \theta $$
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??
$$ \cos\theta = \frac{\pmb{n_1} \cdot \pmb{n_2}}{|\pmb{n_1}||\pmb{n_2}|} $$
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??
$$ \cos\theta = \frac{\pmb{b} \cdot \pmb{n}}{|\pmb{b}||\pmb{n}|} $$
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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date: 2021-01-14 13:15
tags:
- '@?further-maths'
- '@?vectors'
- '@?public'
title: Further Maths - Vector Equation of a Plane
