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Cross Products

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Cross Products

Study Guide

Key Definition

The cross product is an operation that takes two vectors and returns a vector that is perpendicular to both of the input vectors.

Important Notes

The cross product is used to calculate area, torque, and field orientation.
The cross product is anticommutative: $A \times B = - (B \times A)$.
The cross product is distributive: $A \times (B + C) = A \times B + A \times C$.
The scalar multiplication property: $(kA) \times B = A \times (kB) = k(A \times B)$ where $k$ is a scalar.

Mathematical Notation

In the cross product formula, $i$, $j$, and $k$ are the standard unit vectors.

Remember to use proper notation when solving problems

Why It Works

The cross product works by combining the input vectors in a specific formula to produce a new vector that has the unique property of being perpendicular to both input vectors.

Remember

The formula for the cross product is $A \times B = (Ay \cdot Bz - Az \cdot By) i - (Ax \cdot Bz - Az \cdot Bx) j + (Ax \cdot By - Ay \cdot Bx) k$.

Quick Reference

Cross Product:$A \times B = (Ay \cdot Bz - Az \cdot By) i - (Ax \cdot Bz - Az \cdot Bx) j + (Ax \cdot By - Ay \cdot Bx) k$

Understanding Cross Products

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The cross product is a way to multiply two vectors to get another vector. It's used in physics and engineering.

Practice Problems

Test your understanding with practice problems

Scenario Practice

Question Exercise

Beginner

Calculate the Cross Product

Given the vectors A = $(2, 3, 4)$ and B = $(5, 6, 7)$, calculate their cross product.

Practice Quiz

Single Choice Quiz

Intermediate

Which of the following is NOT a property of the cross product?

Advanced Challenge

Thinking Exercise

Advanced

Think About This

Think about how the cross product could be used in physics to calculate the torque on an object.

Recap

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Review key concepts and takeaways