# Further Maths - Cross Product

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/cross-product/ · Updated: 2024-08-24 · Tags: school, further-maths, vectors, further-pure-1

## Flashcards
### 2022-01-25
##### What is the cross product useful for??
Finding a normal vector to two other vectors.

##### What is another name for the cross product??
The vector product.

##### Does the cross product give you a scalar or a vector answer??
A vector.

##### How can you work out the direction of the cross product of two vectors??
Using the right hand rule.

##### In the right hand rule for working out the direction of the cross product, what is the first vector $a$ represented by??
Your first finger.

##### In the right hand rule for working out the direction of the cross product, what is the second vector $b$ represented by??
Your second finger.

##### In the right hand rule for working out the direction of the cross product, what is the resulting normal vector represented by??
Your thumb.

##### Can you demonstrate the right-hand rule by wiggling each finger in turn for $a$, $b$ and $\hat{n}$??
* $a$ is first finger
* $b$ is second finger
* $\hat{n}$ is third finger

##### Is the vector/cross product commutative??
No.

##### What is $\pmb{i} \times \pmb{i}$??
$$
\pmb{0}
$$

##### What is $\pmb{j} \times \pmb{j}$??
$$
\pmb{0}
$$

##### What is $\pmb{k} \times \pmb{k}$??
$$
\pmb{0}
$$

##### What is $\pmb{i} \times \pmb{j}$??
$$
\pmb{k}
$$

##### What is $\pmb{j} \times \pmb{i}$??
$$
\pmb{-k}
$$

##### What is $\pmb{j} \times \pmb{k}$??
$$
\pmb{i}
$$

##### What is $\pmb{k} \times \pmb{j}$??
$$
\pmb{-i}
$$

##### What is $\pmb{k} \times \pmb{i}$??
$$
\pmb{j}
$$

##### What is $\pmb{i} \times \pmb{k}$??
$$
\pmb{-j}
$$

##### What's the nice way of remembering the cross product rules for unit vectors??
* If there are two successive different unit vectors, it's the positive final one.
* If they aren't in order, then it's the negative final one.

##### What's the actual mathematical definition of the cross product in terms of $a$, $b$ and $\theta$??
$$
a \times b = |a||b| \sin \theta \hat{n}
$$

##### What is $$b \times a$$ equivalent to (cross product)??
$$
-a \times b
$$

##### What does it mean for the cross product of two vectors to be $0$??
Either at least one of them is $\pmb{0}$ or they are parallel.

##### How can you use the cross product to work out if two vectors are parallel??
See if it equals $\pmb{0}$.

##### What's the cross product of two vectors $$ \left(\begin{matrix} a_1 \\ a_2 \\ a_3 \end{matrix}\right) \times \left(\begin{matrix} b_1 \\ b_2 \\ b_3 \end{matrix}\right) $$ in terms of the determinant of a matrix??
$$
\left|\begin{matrix} \pmb{i} & \pmb{j} & \pmb{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{matrix}\right|
$$

##### To work out the _unit_ normal vector using the cross product, what must you remember to do by the end??
Divide by the magnitude.

##### What is $\sin \theta$ in terms of $a$ and $b$ using the cross product??
$$
\sin \theta = \frac{|a \times b|}{|a||b|}
$$

##### How could you prove that if $$a + b + c = 0$$ then $$a \times b = b \times c = c \times a$$ (where $a$, $b$ and $c$ are vectors)??
Multiply the original equation by $a$ and then $b$.

##### What's $a \times (b + c)$ (cross product)??
$$
a\times b + a\times c
$$

##### The cross product isn't commutative, but it is gosh-darn rootin'-tootin' what??
Distributive.

### 2022-01-26
##### Why can you take the absolute value of both sides of $a \times b = |a||b|\sin\theta \hat{n}$ to get $|a \times b| = |a||b|\sin\theta$??
Because $|\hat{n}| = 1$

##### If $|a \times b| = |a||b|\sin\theta$ and the area of a triangle is $\frac{1}{2} ab$, then how could you think about $|a \times b|$??
As double the area of a triangle.

##### ![PHOTO TRIANGLE AREA](triangle-area.png) What is the area of this shape in terms of the cross product??
$$
\frac{1}{2} ((c - a) \times (b - a))
$$

##### ![PHOTO PARALLELOGRAM AREA](parallelogram-area.png) What is the area of this shape in terms of the cross product??
$$
(d - a) \times (b - a)
$$

##### What is the scalar triple product for $a$, $b$ and $c$ in terms of the cross product??
$$
a \cdot (b \times c)
$$

##### What is the scalar triple product for $a$, $b$ and $c$ as the determinant of a matrix??
$$
\left|\begin{matrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{matrix}\right|
$$

##### ![PHOTO PARALLELEPIPED AREA](paste-e824e89917800d889f406cb78046ad52148d84a8.jpg) What is the name of this shape??
A parallelepiped.

##### What is the mathematical definition of a parallelepiped??
A 3D shape formed by six parallelograms.

##### ![PHOTO PARALLELEPIPED AREA](parallelepiped.png) What is the formula for the volume of this shape??
$$
|a \cdot (b \times c)|
$$

##### What important property does the scalar triple product have that makes it easier to use??
It is cyclic.

##### If $$\pmb{a} \cdot (\pmb{b} \times \pmb{c}) = d$$ then what does $$\pmb{b} \cdot (\pmb{c} \times \pmb{a})$$ equal??
$$
d
$$

##### If any of the terms in $$\pmb{a} \cdot (\pmb{b} \times \pmb{c})$$ are equal, then what must the result be??
$$
0
$$

##### Does the scalar triple product give a vector or a scalar answer??
A scalar.

##### ![PHOTO TETRAHEDRON VOLUME](tetrahedron-volume.png) What is the name of this shape??
A tetrahedron.

##### ![PHOTO TETRAHEDRON VOLUME](tetrahedron-volume.png) What is the formula for the volume of this shape??
$$
\frac{1}{6} | a \cdot (b \times c) |
$$

##### When using the cross product or scalar triple product to work out volumes, what is important??
That you use direction vectors from a single point, not arbitrary position vectors.

##### How would you prove a shape described with vectors is regular??
Show that all the edge lengths are the same.

### 2022-01-27
#### Straight Lines
##### What is the cross product form of a straight line $r = a + \lambda b$??
$$
(r - a) \times b = 0
$$

##### Why is the cross product form of a straight line useful??
Because it means you don't have to have a parameter.

##### What is another form for the straight line equation $(r - a) \times b = 0$??
$$
r \times b = a \times b
$$

##### What's the geometric interpretation of the cross product straight line formula $(r - a) \times b = 0$??
$r - a$ is the direction vector between a general point and any point on the line, so you're saying that it has to be parallel to the direction vector.

##### For a direction vector $$ a = \left(\begin{matrix} x \\ y \\ z \end{matrix}\right) $$ how can you work out the angle it makes with the $y$-axis??
$$
\cos \theta = \frac{y}{|a|}
$$

##### For a direction vector $$ a = \left(\begin{matrix} x \\ y \\ z \end{matrix}\right) $$ how can you work out the angle it makes with the $x$-axis??
$$
\cos \theta = \frac{x}{|a|}
$$

##### For a direction vector $$ a = \left(\begin{matrix} x \\ y \\ z \end{matrix}\right) $$ how can you work out the angle it makes with the $z$-axis??
$$
\cos \theta = \frac{z}{|a|}
$$

##### What letters are used for the direction cosines of a vector $\frac{x}{|a|}$, $\frac{y}{|a|}$ and $\frac{z}{|a|}$??
$$
l, m, n
$$

##### What is true about the sum of the squares of direction cosines $\frac{x}{|a|}^2$, $\frac{y}{|a|}^2$ and $\frac{z}{|a|}^2$??
They always sum to one.

##### For a vector $$ a = \left(\begin{matrix} x \\ y \\ z \end{matrix}\right) $$ how could you write the unit vector in terms of the direction cosines $\cos \alpha$, $\cos \beta$ and $\cos \gamma$??
$$
\left(\begin{matrix} \cos \alpha \\\\ \cos \beta \\\\ \cos \gamma \end{matrix}\right)
$$

##### How does the cross product make finding the vector equation for a plane easier??
Because it can give you the normal vector to two direction vectors.

##### $$r = a + \lambda b + \mu c$$ What is the dot product form of this plane equation using the cross product??
$$
r \cdot (b \times c) = a \cdot (b \times c)
$$

##### How can you use the cross product to find the line of intersection between two planes??
Find the vector perpendicular to the two normal vectors and then use the "let $z = 0$" approach to finding the point of intersection.

##### If you've got the direction vector of the line of intersection between two planes, how can you find a point that the line goes through given the two plane equations??
Assume that $z = 0$ and find the corresponding $x$ and $y$ values using the two simultaneous equations.

##### $$x + 2y + 3z = 4 \\ 3x + y + 4z = 1$$ You've already found the direction vector for the line of intersection of two planes. How can you now find a point on the plane??
Let $z = 0$ and then solve the equations simultaneously:

$$
x + 2y = 4 \\\\
3x + y = 1
$$

##### What is the formula for the shortest distance between skew lines $$r = a + \lambda b \\ r = c + \mu d$$??
$$
\left| (a - c) \cdot \frac{b \cross d}{|b \times d|} \right|
$$

##### How could you remember the formula for the shortest distance between two skew lines $$r = a + \lambda b \\ r = c + \mu d$$ as a sentence??
It's the vector moving between the two known points ($a - c$) dot producted with the unit vector perpendicular to both direction vectors ($\frac{b\cross d}{|b \cross d|}$)

### 2022-04-18
##### $$ \left( \pmb{r} - \left(\begin{matrix} 2 \\ -1 \\ 2 \end{matrix}\right) \right) \times \left(\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}\right) = \pmb{0}$$ Why do you need to be careful expanding this??
The order of the cross product must be correct otherwise you'll get a negative answer.

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