# Further Maths - Vector Equation of a Plane

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/vector-equation-of-a-plane/ · Updated: 2021-01-14 · Tags: further-maths, vectors

### 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](plane-equation-visual.jpg) 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](scalar-product-plane-form.png) 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](scalar-product-plane-form.png) What does $R$ represent here??
The general position vector of a point on the plane.

##### ![PHOTO SCALAR PRODUCT PLANE FORM](scalar-product-plane-form.png) What does $A$ represent here??
A fixed, known point on the plane.

##### ![PHOTO SCALAR PRODUCT PLANE FORM](scalar-product-plane-form.png) What's the formula for $\overrightarrow{AR}$??
$$
\pmb{r} - \pmb{a}
$$

##### ![PHOTO SCALAR PRODUCT PLANE FORM](scalar-product-plane-form.png) 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](scalar-product-plane-form.png) 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](plane-plane-intersection.jpg) 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](plane-plane-intersection.jpg) 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](plane-line-intersection.jpg) 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](plane-line-intersection.jpg) 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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