Direction of a cross product

Direction of a cross product: linear algebra practice

Develop geometric intuition for the direction of a cross product in three dimensions. Questions use a right-handed coordinate frame, cyclic basis directions and rotations in a coordinate plane. They separate the two possible normal directions, explain how input order chooses one, and handle parallel inputs without assigning a direction to the zero vector.

What you will practise

• Use the right-hand rule in a right-handed coordinate system
• Distinguish perpendicular direction from magnitude and zero cases
• Predict the effects of reversing or rescaling cross-product inputs

How to use this questionset

Prerequisites: Three-dimensional coordinates, perpendicular vectors, angles and the standard right-handed basis. The questions progress from direct interpretation to application and reasoning. Work out an answer before comparing the choices; for multi-select questions, evaluate every statement independently. Worked explanations show the reasoning and address relevant misconceptions, so an incorrect attempt can guide your next review.

All basis examples use i, j and k in a right-handed frame. A zero cross product has no direction; perpendicularity alone does not choose the orientation of a nonzero product. The next microlearning calculates cross-product components and checks the resulting vector algebraically. This microlearning belongs to Module 1, Vectors and Geometry, in the Complete Linear Algebra Course.