Cross products | Chapter 10, Essence of linear algebra

Summary

This video script delves into the concept of cross products, starting with a basic introduction for two-dimensional vectors by visualizing the parallelogram formed and explaining how orientation affects the sign of the cross product. It emphasizes the importance of determinants in calculating the area of this parallelogram, which serves as the magnitude of the cross product. The script then transitions to the true three-dimensional cross product, where a new vector is derived, perpendicular to the original vectors, with magnitude equal to the parallelogram's area and direction determined by the right-hand rule. The video promises a deeper exploration of the cross product's geometric and linear transformational significance in a follow-up video, hinting at the elegance of the underlying mathematical concepts.

Takeaways

• 📚 The video script provides an introduction to the concept of the cross product, with a standard explanation and a deeper understanding in the context of linear transformations.
• 📏 The cross product is initially discussed in two dimensions, involving two vectors and the parallelogram they define.
• 📈 The cross product of two vectors v and w is related to the area of the parallelogram spanned by these vectors, with the sign depending on the orientation (v to the right of w gives a positive area, and vice versa).
• 🔄 The order of the vectors in the cross product matters; swapping them changes the sign of the result.
• 🧭 The orientation can be remembered using the right-hand rule, with the cross product of the basis vectors i-hat and j-hat being positive.
• 🔢 The determinant is used to calculate the cross product in 2D, by placing the vectors as columns in a matrix and computing the determinant.
• 📉 The determinant measures how areas change due to a transformation, which is why it's used to find the area of the parallelogram resulting from the transformation of a unit square.
• 📊 The cross product is not just a number but also a vector in 3D, with its magnitude being the area of the parallelogram and its direction perpendicular to the plane of the two original vectors.
• ✋ The right-hand rule is used to determine the direction of the cross product vector in 3D, ensuring it is perpendicular to both original vectors.
• 🔄 The script mentions a 3D determinant process for calculating the cross product, which involves a matrix with the basis vectors and the vectors being crossed.
• 🤔 The deeper understanding of the cross product in relation to linear transformations and the underlying geometry will be discussed in a follow-up video.

Q & A

What is the cross product of two vectors?

The cross product of two vectors v and w is a vector that is perpendicular to both v and w, with a magnitude equal to the area of the parallelogram defined by v and w. Its direction is determined by the right-hand rule.

Why is the order of vectors important in the cross product?

The order of vectors is important because it determines the orientation of the resulting cross product. If v cross w is positive, then v is on the right of w. If v is on the left of w, then v cross w is negative.

How does the orientation of vectors affect the cross product?

The orientation affects the sign of the cross product. If the vectors are in a right-handed orientation (v on the right of w), the cross product is positive. If they are in a left-handed orientation (v on the left of w), the cross product is negative.

What is the relationship between the cross product and the determinant?

The cross product can be computed using the determinant of a matrix where the columns represent the vectors being crossed. The determinant measures how areas change due to a transformation, and in the case of the cross product, it gives the area of the parallelogram formed by the vectors.

How can you remember the correct order for the cross product of the basis vectors i-hat and j-hat?

You can remember the correct order by noting that the result of i-hat cross j-hat should be positive. Since i-hat is on the right of j-hat, v cross w should be positive when v is on the right of w.

What happens to the cross product if one of the vectors is scaled?

If one of the vectors, say v, is scaled up by a factor of k (e.g., k*v), the cross product is also scaled up by that same factor. So, k*v cross w will be k times the value of v cross w.

What is the geometric interpretation of the cross product vector?

The cross product vector geometrically represents a vector that is perpendicular to the plane containing the original vectors v and w. Its magnitude is equal to the area of the parallelogram spanned by v and w.

How does the right-hand rule help in determining the direction of the cross product?

The right-hand rule helps in determining the direction of the cross product by pointing the forefinger of your right hand in the direction of vector v, and the middle finger in the direction of vector w. The direction of the thumb points in the direction of the cross product.

What is the significance of the parallelogram area in the cross product?

The area of the parallelogram, which is spanned by the two vectors being crossed, is significant because it determines the magnitude of the cross product vector.

How is the cross product computed in three dimensions?

In three dimensions, the cross product is computed using a 3x3 matrix where the first column contains the basis vectors i-hat, j-hat, and k-hat, and the second and third columns contain the coordinates of vectors v and w. The determinant of this matrix gives a linear combination of the basis vectors, which defines the cross product vector.

What is the connection between the cross product and linear transformations?

The cross product is related to linear transformations through the determinant, which measures how areas change due to a transformation. The cross product vector is a result of such a transformation, where the original unit square is transformed into a parallelogram whose area is given by the determinant.

Outlines

00:00

📚 Introduction to Cross Products

This paragraph introduces the concept of cross products in the context of linear transformations, with a plan to explore it in two parts. The first part explains the standard introduction to cross products, starting from two-dimensional vectors and the parallelogram they form. It discusses the orientation and the order of vectors, emphasizing that the cross product is positive when v is on the right of w and negative otherwise. The determinant is introduced as a method to calculate the cross product, relating it to linear transformations and changes in area. An example with specific vector coordinates is used to illustrate the computation of the cross product using determinants. The paragraph concludes with observations about the properties of the cross product, such as its magnitude being larger when vectors are closer to being perpendicular and its behavior under scalar multiplication.

05:02

📐 The True Cross Product in 3D

The second paragraph delves into the true cross product, which is a vector operation in three dimensions. It explains that the cross product results in a vector whose magnitude is the area of the parallelogram defined by the two input vectors, and whose direction is perpendicular to that plane, determined by the right-hand rule. The paragraph provides a concrete example with vectors pointing in the z and y directions, resulting in a cross product pointing in the negative x direction. It also discusses the computation of the cross product using a 3D determinant involving basis vectors. The paragraph hints at a deeper understanding of this computation through the concept of duality, which will be explored in a separate video. The importance of understanding the geometric representation of the cross product vector is emphasized, inviting viewers to delve deeper into the mathematical elegance behind the operation.

Keywords

💡Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In the context of the video, the dot product was previously discussed in relation to linear transformations, serving as a comparison to the cross product which is the main focus of the video's discussion.

💡Cross Product

The cross product is a binary operation on two vectors in a three-dimensional space, resulting in a vector which is perpendicular to the plane containing the two original vectors. It is central to the video's theme, as the script delves into the standard introduction of the concept and a deeper geometric understanding involving linear transformations.

💡Parallelogram

A parallelogram is a four-sided figure with opposite sides that are equal in length and parallel. In the script, the parallelogram is used to visualize the geometric interpretation of the cross product, where the area it spans with two vectors, v and w, is a key component in understanding the magnitude of the cross product.

💡Orientation

Orientation refers to the arrangement or position of an object in space. The script mentions that the cross product takes into account the order of the vectors, which determines the orientation and consequently the sign (positive or negative) of the cross product.

💡Determinant

The determinant is a scalar value that is a function of the entries of a square matrix. It's used in the context of the video to calculate the area of the parallelogram formed by two vectors in 2D space, which is essential for understanding the computation of the cross product.

💡Basis Vectors

Basis vectors are the fundamental building blocks of a vector space. In the script, the basis vectors i-hat and j-hat are used to define the orientation and to compute the cross product through the determinant of a matrix formed by the coordinates of vectors v and w.

💡Right Hand Rule

The right-hand rule is a common mnemonic for understanding the direction of the cross product. The script describes using the right hand to determine the direction of the cross product vector, which is perpendicular to the plane formed by the original vectors v and w.

💡Perpendicular

Perpendicular refers to a line or plane that forms a right angle (90 degrees) with another. The script uses the concept of perpendicularity to explain how the cross product results in a vector that is perpendicular to the plane containing the original vectors.

💡Linear Transformation

A linear transformation is a function that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. The script hints at the deeper connection between linear transformations and the cross product, suggesting a more advanced mathematical concept that will be explored in a follow-up video.

💡Duality

Duality in the context of the video refers to a concept in linear algebra where certain properties or operations have analogous counterparts. The script mentions duality as a concept that will be used to explain the deeper geometric meaning behind the computation of the cross product in a separate follow-on video.

Highlights

Introduction to the concept of the cross product in relation to linear transformations.

Explanation of the cross product in two dimensions using the parallelogram spanned by two vectors v and w.

Cross product's dependence on vector orientation, with v cross w being positive when v is on the right of w.

Importance of order in cross product calculations, where swapping vectors results in a negative value.

Memory aid for cross product ordering using the basis vectors i-hat and j-hat.

Use of determinants for calculating the 2D cross product.

Connection between the determinant, linear transformations, and area changes.

Example calculation of the cross product using determinants with given vector coordinates.

Geometric interpretation of the cross product, including its magnitude and direction.

Demonstration of how the cross product changes with vector scaling.

Introduction to the true 3D cross product, which results in a vector.

Description of the right-hand rule for determining the direction of the cross product vector.

Example using the right-hand rule with perpendicular vectors to find the cross product vector.

Discussion of the 3D determinant process for computing the cross product.

Explanation of the notational trick involving basis vectors in the cross product computation.

Promise of a deeper exploration of the cross product in a follow-up video.

Invitation for curious learners to understand the geometric connection behind the cross product computation.

Transcripts

Last video I talked about the dot product, showing both the standard introduction

to the topic, as well as a deeper view of how it relates to linear transformations.

I'd like to do the same thing for cross products,

which also have a standard introduction, along with a deeper understanding

in the light of linear transformations, but this time I'm dividing it into

two separate videos.

Here, I'll try to hit the main points that students are usually shown

about the cross product, and in the next video I'll be showing a view

which is less commonly taught, but really satisfying when you learn it.

We'll start in two dimensions.

If you have two vectors, v and w, think about the parallelogram that they span out.

What I mean by that is that if you take a copy of v and move its tail to the tip of w,

and you take a copy of w and move its tail to the tip of v,

the four vectors now on the screen enclose a certain parallelogram.

The cross product of v and w, written with the x-shaped multiplication symbol,

is the area of this parallelogram.

Well, almost.

We also need to consider orientation.

Basically, if v is on the right of w, then v cross w

is positive and equal to the area of the parallelogram.

But if v is on the left of w, then the cross product is negative,

namely the negative area of that parallelogram.

Notice this means that order matters.

If you swapped v and w, instead taking w cross v,

the cross product would become the negative of whatever it was before.

The way I always remember the ordering here is that when you take the cross product

of the two basis vectors in order, i-hat cross j-hat, the result should be positive.

In fact, the order of your basis vectors is what defines orientation.

So since i-hat is on the right of j-hat, I remember that v

cross w has to be positive whenever v is on the right of w.

So for example, with the vectors shown here, I'll just

tell you that the area of that parallelogram is seven.

And since v is on the left of w, the cross product should be negative.

So v cross w is negative seven.

But of course, you want to be able to compute this without someone telling you the area.

This is where the determinant comes in.

So if you didn't see chapter five of this series,

where I talk about the determinant, now would be a really good time to go take a look.

Even if you did see it, but it was a while ago,

I'd recommend taking another look just to make sure those ideas are fresh in your mind.

For the 2D cross product, v cross w, what you do is you write the coordinates

of v as the first column of a matrix, and you take the coordinates of w

and make them the second column, then you just compute the determinant.

This is because a matrix whose columns represent v and w corresponds with a

linear transformation that moves the basis vectors i-hat and j-hat to v and w.

The determinant is all about measuring how areas change due to a transformation,

and the prototypical area that we look at is the unit square resting on i-hat and j-hat.

After the transformation, that square gets turned

into the parallelogram that we care about.

So the determinant, which generally measures the factor by which areas are changed,

gives the area of this parallelogram, since it evolved from a square that started with

area one.

What's more, if v is on the left of w, it means that orientation was flipped during

that transformation, which is what it means for the determinant to be negative.

As an example, let's say v has coordinates negative 3, 1, and w has coordinates 2, 1.

The determinant of the matrix with those coordinates as columns

is negative 3 times 1 minus 2 times 1, which is negative 5.

So evidently, the area of the parallelogram they define is 5,

and since v is on the left of w, it should make sense that this value is negative.

As with any new operation you learn, I'd recommend playing

around with this notion a bit in your head, just to get kind

of an intuitive feel for what the cross product is all about.

For example, you might notice that when two vectors are perpendicular,

or at least close to being perpendicular, their cross product is larger than

it would be if they were pointing in very similar directions,

because the area of that parallelogram is larger when the sides are closer to

being perpendicular.

Something else you might notice is that if you were to scale up one of those vectors,

perhaps multiplying v by 3, then the area of that parallelogram

is also scaled up by a factor of 3.

So what this means for the operation is that 3v

cross w will be exactly 3 times the value of v cross w.

Now, even though all of this is a perfectly fine mathematical operation,

what I just described is technically not the cross product.

The true cross product is something that combines

two different 3d vectors to get a new 3d vector.

Just as before, we're still going to consider the parallelogram

defined by the two vectors that we're crossing together,

and the area of this parallelogram is still going to play a big role.

To be concrete, let's say that the area is 2.5 for the vectors shown here.

But as I said, the cross product is not a number, it's a vector.

This new vector's length will be the area of that parallelogram,

which in this case is 2.5, and the direction of that new vector is going to be

perpendicular to the parallelogram.

But which way, right?

I mean, there are two possible vectors with length

2.5 that are perpendicular to a given plane.

This is where the right hand rule comes in.

Point the forefinger of your right hand in the direction of v,

then stick out your middle finger in the direction of w.

Then, when you point up your thumb, that's the direction of the cross product.

For example, let's say that v was a vector with length 2 pointing straight up in

the z direction, and w is a vector with length 2 pointing in the pure y direction.

The parallelogram that they define in this simple example is actually a square,

since they're perpendicular and have the same length, and the area of that square is 4.

So their cross product should be a vector with length 4.

Using the right hand rule, their cross product should point in the negative x direction.

So the cross product of these two vectors is negative 4 times i-hat.

For more general computations, there is a formula that you could memorize if you wanted,

but it's common and easier to instead remember a certain

process involving the 3D determinant.

Now, this process looks truly strange at first.

You write down a 3D matrix where the second and

third columns contain the coordinates of v and w.

But for that first column, you write the basis vectors i-hat, j-hat, and k-hat.

Then you compute the determinant of this matrix.

The silliness is probably clear here.

What on earth does it mean to put in a vector as the entry of a matrix?

Students are often told that this is just a notational trick.

When you carry out the computations as if i-hat, j-hat, and k-hat were numbers,

then you get some linear combination of those basis vectors.

And the vector defined by that linear combination, students are told to just believe,

is the unique vector perpendicular to v and w,

whose magnitude is the area of the appropriate parallelogram,

and whose direction obeys the right hand rule.

And sure, in some sense this is just a notational trick,

but there is a reason for doing it.

It's not just a coincidence that the determinant is once again important.

And putting the basis vectors in those slots is not just a random thing to do.

To understand where all of this comes from, it helps to

use the idea of duality that I introduced in the last video.

This concept is a little bit heavy though, so I'm putting it in a

separate follow-on video for any of you who are curious to learn more.

Arguably, it falls outside the essence of linear algebra.

The important part here is to know what that cross

product vector geometrically represents.

So if you want to skip that next video, feel free.

But for those of you who are willing to go a bit deeper,

and who are curious about the connection between this computation and the underlying

geometry, the ideas that I'll talk about in the next video are just a really

elegant piece of math.