Further Maths - Vector Equation of a Plane
2021-01-14
What is the general vector equation of a plane?
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\[\pmb{r} = \pmb{a} = \lambda\pmb{b} + \mu\pmb{c}\]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?
What equation does this photo represent?#####
\[\left(\begin{matrix} 3+2\lambda+\mu \\ 4+\lambda-\mu \\ -2+\lambda+2\mu \end{matrix}\right) = \left(\begin{matrix} 2 \\ 2 \\ -1 \end{matrix}\right)\]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.
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\[\left(\begin{matrix} 2\lambda+\mu \\\\ \lambda-\mu \\\\ \lambda+2\mu \end{matrix}\right) = \left(\begin{matrix} -1 \\\\ -2 \\\\ 1 \end{matrix}\right)\]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?
What is the general form of the Cartesian equation of a plane?
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.
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\[2x + 3y + 5z = 5\]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?
What does the parametric equation of a plane look like?
What does the scalar product equation of a line look like?
What are the three types of plane equation?
- Cartesian
- Parametric
- Scalar product
What does $\pmb{n}$ represent here?
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?
What does $R$ represent here?The general position vector of a point on the plane.
What does $A$ represent here?
What does $A$ represent here?A fixed, known point on the plane.
What’s the formula for $\overrightarrow{AR}$?
What’s the formula for $\overrightarrow{AR}$? What’s true about the line $\pmb{r} - \pmb{a}$ in relation to the normal vector $\pmb{n}$?
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 $\pmb{n}$?
How would you write $\pmb{r} - \pmb{a}$ being perpindicular to the normal vector $\pmb{n}$?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.
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\[\pmb{r}\cdot\pmb{n} = d\]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?
If the angle between the normals of two intersecting lines is $\theta$, what is the angle between the two intersecting planes? If the two normals are $\pmb{n _ 1}$ and $\pmb{n _ 2}$, what’s the formula for $\cos\theta$?
If the two normals are $\pmb{n _ 1}$ and $\pmb{n _ 2}$, what’s the formula for $\cos\theta$?#####
\[\pmb{r}\cdot\pmb{n _ 1} = k _ 1 \\ \pmb{r}\cdot\pmb{n _ 2} = k _ 2\]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?
If the angle between the line and the normal to the plane is $\theta$, what is the angle between the plane and the line?#####
\[\pmb{r}\cdot\pmb{n} = k \\ \pmb{r} = \pmb{a} + \lambda\pmb{b}\]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?
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?
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.