The Vector Cross Product

Summary

In this lesson, Professor Dave explains the concept of the cross product, a method of multiplying two vectors to get a new vector that is perpendicular to both. By using determinants of a 3x3 matrix, he demonstrates how to find the cross product of two vectors, A and B. He covers key points like the right-hand rule for direction, how the magnitude of the cross product relates to the sine of the angle between the vectors, and that the cross product of parallel vectors is zero. The properties of the cross product, such as non-commutativity and distributivity, are also discussed.

Takeaways

• 🧮 The cross product is a way to multiply two vectors to get another vector, unlike the dot product, which results in a scalar.
• 📐 The cross product of two vectors A and B is calculated using the determinant of a 3x3 matrix involving the unit vectors I, J, K, and the components of A and B.
• 🖋️ The resulting vector from the cross product is orthogonal (perpendicular) to both original vectors or the plane they form.
• 👍 The right-hand rule helps determine the direction of the cross product: curling your fingers from B to A makes your thumb point in the direction of the cross product.
• 🚫 The cross product of a vector with itself is always zero, which can be proven by calculating the determinant.
• 📏 The magnitude of the cross product is equal to the product of the magnitudes of the two vectors multiplied by the sine of the angle between them.
• 🔄 The cross product is not commutative: A cross B equals negative B cross A.
• 🚫 The cross product is not associative: (A cross B) cross C is not the same as A cross (B cross C).
• ➕ The cross product is distributive: A cross (B + C) equals A cross B plus A cross C.
• 🔺 The magnitude of the cross product represents the area of the parallelogram formed by the two vectors, and the cross product of parallel vectors is zero since they span no area.

Q & A

What is the cross product, and how does it differ from the dot product?

The cross product is a way to multiply two vectors to get another vector, while the dot product results in a scalar. The cross product yields a vector that is perpendicular to the two original vectors.

How do you find the cross product of two vectors?

To find the cross product of two vectors, you calculate the determinant of a 3x3 matrix with the unit vectors (i, j, k) in the top row, the components of the first vector (A) in the second row, and the components of the second vector (B) in the third row.

What is the right-hand rule, and how does it relate to the cross product?

The right-hand rule helps determine the direction of the cross product. If you place your right hand on vector B and curl your fingers toward vector A, your thumb points in the direction of the cross product.

Why is the cross product of two parallel vectors equal to zero?

The cross product of two parallel vectors is zero because the angle between them is zero, and the sine of zero is zero. As a result, the magnitude of the cross product becomes zero.

What does the magnitude of the cross product represent?

The magnitude of the cross product represents the product of the magnitudes of the two vectors and the sine of the angle between them. It also corresponds to the area of the parallelogram formed by the two vectors.

What happens when you take the cross product of a vector with itself?

The cross product of a vector with itself is always zero. This is because the sine of the angle between the same vector is zero, leading to a zero cross product.

Is the cross product commutative? Explain.

No, the cross product is not commutative. A cross B is not equal to B cross A. Instead, A cross B equals the negative of B cross A.

Is the cross product associative?

No, the cross product is not associative. The expression (A cross B) cross C is not equal to A cross (B cross C).

What is the relationship between the cross product and the area of a parallelogram?

The magnitude of the cross product of two vectors gives the area of the parallelogram formed by those vectors. This is because the magnitude is the product of the vectors' magnitudes and the sine of the angle between them.

Does the cross product distribute over vector addition?

Yes, the cross product distributes over vector addition. A cross (B + C) is equal to A cross B plus A cross C.

Outlines

00:00

🧑‍🏫 Understanding Cross Products in Vector Algebra

Professor Dave introduces the concept of the cross product, a method of multiplying two vectors to yield a new vector, unlike the dot product, which gives a scalar. He starts by recapping vector operations, mentioning the dot product and its scalar result, before explaining that the cross product yields a vector. He then walks through an example with two vectors, A and B, using their components to form a 3x3 matrix with unit vectors i, j, and k. By calculating the determinant of this matrix, he demonstrates how to obtain the cross product, resulting in a new vector that is orthogonal (perpendicular) to the original two vectors. This perpendicularity is key to understanding the behavior of the cross product in three-dimensional space.

05:03

👋 Applying the Right-Hand Rule to Determine Cross Product Direction

The right-hand rule helps visualize the direction of the cross product. Dave explains how the orientation of the fingers can mimic the angle between vectors A and B, with the thumb pointing in the direction of their cross product. Another variation of this rule involves pointing the index finger along vector A, the middle finger along vector B, and the thumb will indicate the cross product direction. This method is crucial for determining the orientation of the resulting vector. In addition, he emphasizes that the cross product of a vector with itself is zero, and he introduces the formula for the magnitude of the cross product, which is tied to the sine of the angle between the two vectors.

🟰 Parallel Vectors and Zero Cross Product

Dave further explains the relationship between cross products and vector properties. When two vectors are parallel, their cross product is zero, as the sine of the angle between them (0°) equals zero. This also explains why parallel vectors don't form a parallelogram with area—because the area spanned by the vectors is zero. He ties the cross product back to geometry, noting that the magnitude of the cross product vector represents the area of the parallelogram formed by the two vectors.

🚫 Non-Commutative and Non-Associative Properties of Cross Products

The cross product has specific algebraic properties, notably being non-commutative and non-associative. Dave explains that A cross B is not the same as B cross A; instead, A cross B equals the negative of B cross A. Additionally, the cross product does not follow the associative law, meaning (A cross B) cross C is not the same as A cross (B cross C). However, it is distributive, as A cross (B + C) equals A cross B plus A cross C. He concludes by noting that the cross product is significant in both physics and linear algebra, encouraging further exploration of the topic.

Keywords

💡Cross Product

The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space. It results in a third vector which is perpendicular to the plane containing the two input vectors. In the script, the cross product of vectors A and B is calculated by finding the determinant of a matrix formed by unit vectors and the components of A and B. The cross product is central to the video's theme as it introduces a method to derive a new vector from two existing vectors, which is crucial in various applications of linear algebra and physics.

💡Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It is mentioned in the script as a contrasting operation to the cross product. While the dot product results in a scalar, the cross product results in a vector. The script uses the dot product to highlight the difference in the outcome of vector operations.

💡Determinant

The determinant is a scalar value that can be computed from a matrix and is used extensively in linear algebra. In the context of the video, the determinant is used to calculate the cross product by constructing a 3x3 matrix with the unit vectors and the components of vectors A and B. The determinant is key to understanding how the components of the resulting vector from the cross product are found.

💡Unit Vectors

Unit vectors are vectors of length or magnitude 1. Often denoted by i, j, and k for the x, y, and z axes respectively, they are used in the script to represent the direction of the cross product. They are part of the matrix setup for calculating the cross product and are essential in expressing the outcome as a vector.

💡Orthogonal

Orthogonal vectors are those that are perpendicular to each other. The script explains that the cross product of two vectors always results in a vector that is orthogonal to the original vectors. This property is important in understanding the geometric interpretation of the cross product.

💡Right-Hand Rule

The right-hand rule is a common mnemonic for understanding the orientation of a cross product. By aligning the right hand with one vector and curling the fingers towards the second vector, the thumb points in the direction of the cross product. The script uses the right-hand rule to provide a practical way to determine the direction of the cross product.

💡Magnitude

The magnitude of a vector is its length or size. In the script, the magnitude of the cross product is described as the product of the magnitudes of the two original vectors and the sine of the angle between them. This concept is used to understand the length of the resulting vector from the cross product.

💡Parallel Vectors

Parallel vectors are vectors that are scalar multiples of each other and thus point in the same or opposite directions. The script mentions that the cross product of parallel vectors is zero because the angle between them is zero, and the sine of zero is zero.

💡Commutative Property

The commutative property states that the order of operations does not affect the result. The script clarifies that the cross product does not have this property; the cross product of A and B is not the same as the cross product of B and A, which is a critical distinction in vector algebra.

💡Associative Property

The associative property allows one to group operations in a different way without changing the result. The script notes that the cross product is not associative, meaning that (A cross B) cross C is not equal to A cross (B cross C), which is an important property to remember when performing multiple cross products.

💡Distributive Property

The distributive property allows one to distribute multiplication over addition. The script mentions that the cross product is distributive over vector addition, meaning A cross (B + C) equals A cross B + A cross C. This property is useful for simplifying calculations involving multiple vectors.

Highlights

Introduction to vectors and the cross product as a way to multiply two vectors to get a new vector.

Cross product is denoted by the multiplication sign and results in a vector, unlike the dot product which results in a scalar.

Example of cross product between vectors A (1, 3, 4) and B (2, 7, -5), involving the determinant of a 3x3 matrix.

The process of calculating the cross product involves finding the determinant of 2x2 matrices for each unit vector (I, J, K).

Explanation of the right-hand rule to determine the direction of the cross product.

The cross product of two vectors will always yield a vector that is perpendicular (orthogonal) to the original vectors.

If you take the cross product of a vector with itself, the result will always be the zero vector.

The magnitude of the cross product is the product of the magnitudes of the two vectors times the sine of the angle between them.

Cross product of parallel vectors is always zero, since the sine of the angle between them (zero degrees) is zero.

The magnitude of the resulting vector from a cross product represents the area of the parallelogram created by the two vectors.

The cross product is not commutative; A cross B equals negative B cross A.

The cross product is not associative, meaning (A cross B) cross C is not equal to A cross (B cross C).

The cross product distributes over addition: A cross (B + C) equals A cross B plus A cross C.

The cross product is frequently used in physics and other applications of linear algebra.

Understanding the cross product and its properties is essential for applications in linear algebra and physics.

Transcripts

Professor Dave here, let’s find cross products.

Earlier in the series, we learned about vectors and their operations, including one way to

multiply two vectors together, which we called the dot product.

But the dot product is just a scalar.

There is another way to multiply two vectors to get another vector, and this new vector

is called the cross product, which is denoted by the multiplication sign.

Now that we know a little bit about matrices and their operations, we are ready to understand

how to find the cross product of two vectors, so let’s learn how to do that now.

Let’s take two vectors, A and B. A has components one, three, four, which can also be expressed

as I plus three J plus four K, and B has components two, seven, negative five, which can be expressed

as two I plus seven J minus five K.

The way we find this cross product, or A cross B, is by finding the determinant of the following

three by three matrix, with I, J, and K across the top row, the components from vector A

in the second row, and the components from vector B in the third row.

We know how to get the determinant of a three by three matrix, so this is no problem.

We just get I times the determinant of this two by two matrix, minus J times the determinant

of this one, plus K times the determinant of this one.

Now we can find each of these determinants.

First we have negative fifteen minus twenty eight, then negative five minus eight, and

then seven minus six.

Simplifying and combining with the unit vectors, we are left with negative forty three I, plus

thirteen J, plus K.

So we can see how using determinants, we have found the cross product of these two vectors,

which is another, different vector.

We must understand that a vector cross product will always yield a vector that is orthogonal,

or perpendicular, to the original two vectors, or rather to the plane containing them.

In fact, if you place the edge of your right hand directly on vector B and curl your fingers

in the direction of vector A, so that your hand is sort of mimicking the angle the angle

formed between them, your thumb will be pointing in the direction of their cross product.

Alternately, you can point your index finger in the direction of A, and your middle finger

in the direction of B, and again your thumb will be pointed in the direction of the cross product.

So the right-hand rule that we originally learned for three-dimensional coordinate systems

can help us understand the direction of a cross product.

There are a few other key points to understand about cross products.

First, if you take the cross product of a vector and itself, like A cross A, you will

always get zero.

This will be easy to prove to yourself if you work out the determinant just as we did

before and see that all the terms cancel out.

Also, we can make the following statement about the length of the cross product vector.

The magnitude of A cross B is equal to the magnitude of A times the magnitude of B times

the sine of the angle between them.

This definition can also be used to find the cross product of two vectors, because magnitude

and direction are the two pieces of information conveyed by any vector, and if we calculate

the magnitude of the cross product this way, and then determine its direction by using

the right-hand rule, we can therefore accurately define this cross product.

From this method we can also arrive at another truth, that any two parallel vectors must

have a cross product equal to zero.

That’s because parallel vectors have an angle of zero between them, and the sine of

zero is zero, so the whole cross product, according to this expression, must go to zero.

One other interesting thing about the cross product is that the magnitude of the resulting

vector, which is the AB sine theta expression we just used, will also be equal to the area

of the parallelogram created by vectors A and B.

This is another reason that the cross product of parallel vectors is the zero vector, because

they span zero area.

Lastly, just as we learned some properties of the dot product, let’s quickly mention

some properties of the cross product.

First, the cross product is not commutative.

A cross B does not equal B cross A. Rather, A cross B equals negative B cross A. Second,

the cross product is not associative.

The quantity A cross B, cross C is not equal to A cross the quantity B cross C. However,

the cross product does distribute.

A cross the quantity B plus C is equal to A cross B plus A cross C.

This understanding of the cross product will suffice for our purposes, and we should note

that the cross product crops up all the time in physics, but let’s keep going with linear

algebra, right after we check comprehension.