Product of Vectors

Chapter 2 — Vectors · Topic: Product of Vectors Subtopics: 1. Scalar Product or Dot Product 2. Vector Product or Cross Product

Learning Outcomes

• Explain the meaning of vector multiplication.
• Differentiate between dot product and cross product.
• Calculate the dot product using magnitudes and components.
• Calculate the angle between two vectors using their dot product.
• Calculate the magnitude and direction of a cross product.
• Apply the right-hand rule correctly.
• Calculate a cross product using Cartesian components.
• Apply vector products to work, torque, power, angular momentum, and magnetic force.

Definition of Product of Vectors

Dot Product and Cross Product Comparison

Dot product versus cross product

Key Logic

Tips and Tricks

Tips and Tricks

• Check whether the required answer is scalar or vector.
• Use cosine for the dot product.
• Use sine for the cross product.
• Use the smaller angle between the vectors.
• Never attach a direction to a dot-product answer.
• Always determine the direction of a cross product.
• Changing the order does not change the dot product.
• Changing the order reverses the cross product.
• Check the signs carefully in component calculations.
• Keep calculator mode in degrees when angles are given in degrees.

Common Mistakes

Common Mistakes

• Using sine in the dot-product formula.
• Using cosine in the cross-product formula.
• Writing a vector as the answer to a dot product.
• Forgetting the direction of a cross product.
• Assuming the cross product is commutative.
• Using the wrong order in unit-vector multiplication.
• Missing the negative sign in the ĵ term of determinant expansion.
• Confusing A⃗ · A⃗ with A⃗ × A⃗.
• Forgetting that perpendicular vectors have zero dot product.
• Forgetting that parallel vectors have zero cross product.

Topic Summary

Vectors can be multiplied using scalar and vector products. The dot product produces a scalar and measures the parallel relationship between vectors. The cross product produces a perpendicular vector and measures the perpendicular interaction between vectors. Correct formulas, component signs, vector order, angles, and direction rules are essential for solving these problems.