The cross product is a way to multiply two vectors in three dimensions and get a new vector. Unlike the dot product, which gives a scalar, the cross product gives a vector that points perpendicular to both original vectors. It is important in geometry, physics, engineering, and computer graphics because it connects direction, rotation, and area.
The direction of a × b is found using the right-hand rule, so order matters.
Key Facts
- The cross product a × b is a vector perpendicular to both a and b.
- Magnitude formula: |a × b| = |a||b|sin(theta), where theta is the angle between a and b.
- Component formula: a × b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1).
- Determinant form: a × b = det[[i, j, k], [a1, a2, a3], [b1, b2, b3]].
- Order matters: a × b = -(b × a).
- If a and b are parallel, then a × b = 0 because sin(0) = 0 or sin(180°) = 0.
Vocabulary
- Cross product
- A vector operation on two 3D vectors that produces a vector perpendicular to both inputs.
- Right-hand rule
- A method for finding the direction of a cross product by curling the fingers of the right hand from the first vector toward the second vector.
- Magnitude
- The length or size of a vector, written with absolute value bars such as |a|.
- Unit vector
- A vector with length 1 that is often used to describe direction.
- Determinant
- A number calculated from a square array that can be used to write the cross product in component form.
Common Mistakes to Avoid
- Reversing the order of the vectors, then keeping the same answer. This is wrong because a × b and b × a point in opposite directions.
- Using cos(theta) instead of sin(theta) for the magnitude. The cross product measures perpendicular spread and area, so |a × b| = |a||b|sin(theta).
- Forgetting the negative sign in the j component of the determinant. This changes the direction and gives an incorrect vector.
- Treating the cross product like ordinary multiplication of components. The cross product combines components in a specific pattern and produces a perpendicular vector, not (a1b1, a2b2, a3b3).
Practice Questions
- 1 Compute a × b for a = (2, 3, 1) and b = (4, 0, 5).
- 2 Two vectors have lengths |a| = 6 and |b| = 10, and the angle between them is 30°. Find |a × b|.
- 3 A vector a points along the positive x-axis and vector b points along the positive y-axis. Use the right-hand rule to determine the direction of a × b, then explain what changes if the order is reversed.