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.
\[\pmb{r} = \pmb{a} = \lambda\pmb{b} + \mu\pmb{c}\]
What must be true about the two directional vectors $\pmb{b}$ and $\pmb{c}$?
They are not parallel to one another.
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\[\left(\begin{matrix} 3+2\lambda+\mu \\ 4+\lambda-\mu \\ -2+\lambda+2\mu \end{matrix}\right) = \left(\begin{matrix} 2 \\ 2 \\ -1 \end{matrix}\right)\]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.
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?
\[\left(\begin{matrix} 2\lambda+\mu \\\\ \lambda-\mu \\\\ \lambda+2\mu \end{matrix}\right) = \left(\begin{matrix} -1 \\\\ -2 \\\\ 1 \end{matrix}\right)\]
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?
- $\overrightarrow{AB}$
- $\overrightarrow{AC}$
- $\overrightarrow{BC}$
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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??
\[n_1x + n_2y + n_3z\]If there are three points $A, B, C$ on a plane, how could you write the plane equation?
2021-01-18
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.
2021-01-20
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?
\[\left(\begin{matrix} n _ 1 \\ n _ 2 \\ n _ 3 \end{matrix}\right)\]
What’s the Cartesian equation of the plane if $n$ is the normal vector?
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.
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.
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.
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??
\[\left(\begin{matrix} x \\\\ y \\\\ z \end{matrix}\right) \cdot \pmb{n} = d\]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.
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\[\pmb{r}\cdot\pmb{n _ 1} = k _ 1 \\ \pmb{r}\cdot\pmb{n _ 2} = k _ 2\]What is the formula for $cos\theta$, the angle between the two intersecting planes??
\[\cos\theta = \frac{\pmb{n_1} \cdot \pmb{n_2}}{|\pmb{n_1}||\pmb{n_2}|}\] What does $A$ represent here?
What does $A$ represent here?A fixed, known point on the plane.
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\[\pmb{r}\cdot\pmb{n} = k \\ \pmb{r} = \pmb{a} + \lambda\pmb{b}\]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??
\[\cos\theta = \frac{\pmb{b} \cdot \pmb{n}}{|\pmb{b}||\pmb{n}|}\] What’s the formula for $\overrightarrow{AR}$?
What’s the formula for $\overrightarrow{AR}$?2021-01-22
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.
2021-05-17
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 $\pmb{n}$?