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.