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Math 126

Lecture 4

Behind us

Questions?

Today

Cross products and the parallelepipeds they love

Read Section 12.4 of the book. We will cover everything in lecture or section. That's OK.

Warm up

Question: two vectors span a parallelogram.

• What is the area of the parallelogram?
• What is a unit vector perpendicular to the parallelogram?

Warm up

Example: how about the parallelogram spanned by the two vectors $\langle 4,0\rangle$ and $\langle 3,4\rangle$ in the $xy$-plane?

Warm up

Example: how about the parallelogram spanned by the two vectors $\langle 4,0\rangle$ and $\langle 3,4\rangle$ in the $xy$-plane?

Warm up

Example: how about the parallelogram spanned by the two vectors $\langle 4,0\rangle$ and $\langle 3,4\rangle$ in the $xy$-plane?

The area is thus $4\cdot 4=16$.

Magic on the way

It turns out that there is a way to make a single vector encoding both the perpendicular direction and area of the parallelogram.

Magic on the way

It turns out that there is a way to make a single vector encoding both the perpendicular direction and area of the parallelogram.

Secret sauce: the cross product

Definition

The cross product of two vectors $$\mathbf a=\langle a_1,a_2,a_3\rangle\textrm{ }\mathbf b=\langle b_1,b_2,b_3\rangle$$ is the vector $$\begin{align*} \mathbf a&\times\mathbf b=\\ &\langle a_2b_3-a_3b_2, a_3b_1-a_1b_3, a_1b_2-a_2b_1\rangle. \end{align*}$$

Definition

The cross product of two vectors $$\mathbf a=\langle a_1,a_2,a_3\rangle\textrm{ }\mathbf b=\langle b_1,b_2,b_3\rangle$$ is the vector $$\begin{align*} \mathbf a&\times\mathbf b=\\ &\langle a_2b_3-a_3b_2, a_3b_1-a_1b_3, a_1b_2-a_2b_1\rangle. \end{align*}$$

Note: unlike the dot product, this is a vector!

Computing with a determinant

This is a slight abuse of notation, but we have $$\mathbf a\times\mathbf b=\left|\begin{array}{ccc} \mathbf i & \mathbf j & \mathbf k\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\end{array}\right|$$

Computing with a determinant

This is a slight abuse of notation, but we have $$\mathbf a\times\mathbf b=\left|\begin{array}{ccc} \mathbf i & \mathbf j & \mathbf k\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\end{array}\right|$$

You read about how to compute this in the book. Let's review it!

Computing with a determinant

$$ \begin{align*} \mathbf a \times\mathbf b&=\left|\begin{array}{ccc} \mathbf i & \mathbf j & \mathbf k\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\end{array}\right|\\&\\&=\mathbf i\left|\begin{array}{cc}a_2 & a_3\\ b_2 & b_3\end{array}\right|- \mathbf j\left|\begin{array}{cc}a_1 & a_3\\ b_1 & b_3\end{array}\right|+ \mathbf k\left|\begin{array}{cc}a_1 & a_2\\ b_1 & b_2\end{array}\right| \end{align*}$$

Computing with a determinant

$$ \begin{align*} \mathbf a \times\mathbf b&=\left|\begin{array}{ccc} \mathbf i & \mathbf j & \mathbf k\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\end{array}\right|\\&\\&=\mathbf i\left|\begin{array}{cc}a_2 & a_3\\ b_2 & b_3\end{array}\right|- \mathbf j\left|\begin{array}{cc}a_1 & a_3\\ b_1 & b_3\end{array}\right|+ \mathbf k\left|\begin{array}{cc}a_1 & a_2\\ b_1 & b_2\end{array}\right|\\ &=\mathbf i(a_2b_3-a_3b_2)-\mathbf j(a_1b_3-a_3b_1)\\ &\quad\quad\quad+\mathbf k(a_1b_2-a_2b_1) \end{align*}$$

Example

$$\begin{align*} \langle 1,1,0\rangle\times&\langle 0,1,1\rangle\\ =&\left|\begin{array}{ccc} \mathbf i & \mathbf j & \mathbf k\\ 1 & 1 & 0\\ 0 & 1 & 1\end{array}\right|\\ \end{align*}$$

Example

$$\begin{align*} \langle 1,1,0\rangle\times&\langle 0,1,1\rangle\\ =&\left|\begin{array}{ccc} \mathbf i & \mathbf j & \mathbf k\\ 1 & 1 & 0\\ 0 & 1 & 1\end{array}\right|\\ =&\mathbf i\left|\begin{array}{cc}1 & 0\\ 1 & 1\end{array}\right|- \mathbf j\left|\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right|+ \mathbf k\left|\begin{array}{cc}1 & 1\\ 0 & 1\end{array}\right| \end{align*}$$

Example

$$\begin{align*} \langle 1,1,0\rangle\times&\langle 0,1,1\rangle\\ =&\left|\begin{array}{ccc} \mathbf i & \mathbf j & \mathbf k\\ 1 & 1 & 0\\ 0 & 1 & 1\end{array}\right|\\ =&\mathbf i\left|\begin{array}{cc}1 & 0\\ 1 & 1\end{array}\right|- \mathbf j\left|\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right|+ \mathbf k\left|\begin{array}{cc}1 & 1\\ 0 & 1\end{array}\right|\\ =&\mathbf i-\mathbf j+\mathbf k=\langle 1,-1,1\rangle\\ \end{align*}$$

Brain squeeze

Compute these cross products and draw the resulting vectors:

• $\langle 1,2,3\rangle\times\langle 3,6,9\rangle$
• $\mathbf i\times\mathbf i$
• $\mathbf i\times\mathbf j$
• $\mathbf j\times\mathbf i$
• $\mathbf j\times \mathbf k$
• $\mathbf k\times \mathbf i$

Sweet Theorem, Cross Product Version

Given two vectors $\mathbf a$ and $\mathbf b$ with angle $\theta$

• the vector $\mathbf a\times\mathbf b$ is $\perp$ to both $\mathbf a$ and $\mathbf b$;
• the list $\mathbf a,\mathbf b,\mathbf a\times\mathbf b$ satisfies the right-hand rule;
• we have $|\mathbf a\times\mathbf b|=|\mathbf a||\mathbf b|\sin(\theta)$;
• $|\mathbf a\times\mathbf b|$ equals the area of the parallelogram spanned by $\mathbf a$ and $\mathbf b$.

Sweet Sweet Theorem, Cross Product Version

Given two vectors $\mathbf a$ and $\mathbf b$ with angle $\theta$

• the vector $\mathbf a\times\mathbf b$ is $\perp$ to both $\mathbf a$ and $\mathbf b$;
• the list $\mathbf a,\mathbf b,\mathbf a\times\mathbf b$ satisfies the right-hand rule;
• we have $|\mathbf a\times\mathbf b|=|\mathbf a||\mathbf b|\sin(\theta)$;
• $|\mathbf a\times\mathbf b|$ equals the area of the parallelogram spanned by $\mathbf a$ and $\mathbf b$.

Ties the cross product to area and orientation .

Do one

Example: find the area of the parallelogram spanned by $\langle 4,0,0\rangle$ and $\langle 3,4,0\rangle$

Compute the cross product!

• Can you predict the direction of the cross product vector without calculating anything?
• Can you predict the magnitude of the cross product without calculating anything?

I did it

Example: find the area of the parallelogram spanned by $\langle 4,0,0\rangle$ and $\langle 3,4,0\rangle$

Compute the cross product!

$\langle 4,0,0\rangle\times\langle 3,4,0\rangle=\langle 0,0,16\rangle$

Phew!

Hmmm.... Maybe drawing a picture of this (the parallelogram and the cross product) would help you cement this in your mind.

Exploratory probe

Suppose you model a surface with a computer: you can't do continuous or smooth things, so the surface is modeled as having a ton of faces (like the facets of a demonic gem). These faces might be described with coordinates in $\mathbf R^3$. How can you find the normal vector to one of the faces?

How many ways are there to make a normal vector? Is it unique? Can you use them to distinguish "inside" from "outside"?

Use the internet to find out if these things come up outside of math class!

Next time: lines and planes!