The Vector Cross Product
Summary
TLDRIn 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
🧑🏫 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.
👋 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.
Mindmap
Keywords
💡Cross Product
💡Dot Product
💡Determinant
💡Unit Vectors
💡Orthogonal
💡Right-Hand Rule
💡Magnitude
💡Parallel Vectors
💡Commutative Property
💡Associative Property
💡Distributive Property
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.
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