The Vector Cross Product
TLDRThe video explains the cross product, a way to multiply two vectors to yield another vector perpendicular to the original two. Using determinants of matrices containing the vector components, the cross product's direction complies with the right-hand rule. Key properties: taking the cross product of a vector with itself yields zero; its magnitude equals the product of the operand magnitudes times the sine of their included angle; it is anticommutative and does not generally associate. The cross product finds extensive application in physics.
Takeaways
- 😀 The cross product is a way to multiply two vectors to get another vector, denoted by the multiplication sign.
- 👉 To find the cross product, put the components of the vectors into a 3x3 matrix with I, J, K as the top row and take the determinant.
- ✋ The right hand rule shows the direction of the cross product vector relative to the two input vectors.
- 👌 The cross product of a vector with itself is always zero.
- 🤏 The magnitude of the cross product equals the product of the input vector magnitudes and sine of their angle.
- 😕 The cross product is not commutative - order matters.
- 😟 The cross product is not associative.
- 🙂 The cross product distributes over vector addition.
- 🧐 Parallel vectors have a cross product of zero.
- 🤓 The area of the parallelogram spanned by two vectors equals the magnitude of their cross product.
Q & A
What is the cross product of two vectors?
-The cross product of two vectors results in another vector that is perpendicular or orthogonal to the plane containing the original two vectors.
How do you calculate the cross product of two vectors?
-To calculate the cross product of two vectors A and B, you take the determinant of a 3x3 matrix with the unit vectors I, J and K across the top row, the components of vector A in the second row, and the components of vector B in the third row.
How can you use the right-hand rule to find the direction of a cross product?
-Place the edge of your right hand on vector B and curl your fingers towards vector A. Your thumb will then point in the direction of the cross product vector.
What is the result when you take the cross product of a vector with itself?
-Taking the cross product of a vector with itself, such as A x A, will always result in the zero vector.
How do you calculate the magnitude of a cross product?
-The magnitude of the cross product A x B is equal to the magnitude of A times the magnitude of B times the sine of the angle between them.
Why is the cross product of two parallel vectors equal to the zero vector?
-Two parallel vectors have an angle of 0 degrees between them. Since sine of 0 is 0, the magnitude of their cross product, which involves sine of the angle, is 0. Therefore, the cross product is the 0 vector.
What is the relationship between a cross product and parallelogram area?
-The magnitude of a cross product of two vectors is equal to the area of the parallelogram created by those two vectors.
Is the cross product commutative? Why or why not?
-No, the cross product is not commutative. A x B does not equal B x A. Instead, A x B equals negative B x A.
Is the cross product associative? Why or why not?
-No, the cross product is not associative. The quantity A x (B x C) does not equal (A x B) x C.
Does the cross product obey the distributive property?
-Yes, the cross product does obey the distributive property. A x (B + C) equals (A x B) + (A x C).
Outlines
😀 Understanding Cross Products of Vectors
This paragraph introduces the concept of the cross product of two vectors. It defines the cross product, shows how to calculate it using determinants, and explains geometrically that the cross product vector is perpendicular to the plane containing the original vectors. It also introduces the right-hand rule for finding the direction of the cross product.
😀 Key Properties and Uses of Cross Products
This paragraph highlights key properties and applications of cross products. It notes that a vector crossed with itself is zero, gives the formula for the magnitude of the cross product, and explains how cross products relate to areas of parallelograms. It also lists some key differences from dot products regarding commutativity and associativity.
Mindmap
Keywords
💡vector
💡cross product
💡determinant
💡orthogonal
💡right-hand rule
💡parallel vectors
💡area of parallelogram
💡properties
💡commutative
💡associative
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Transcripts
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