# 14: Vectors

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##### Objectives
• Be able to perform operations with vectors: addition, subtraction, dot product, cross product.
• Understand how to calculate the modulus of a vector, including vectors containing complex entries.
• Understand how to normalize vectors.

• 14.1: Introduction to Vectors
A vector is a quantity that has both a magnitude and a direction, and as such they are used to specify the position, velocity and momentum of a particle, or to specify a force.
• 14.2: The Scalar Product
The scalar product of vectors u and v , also known as the dot product or inner product, is defined as (notice the dot between the symbols representing the vectors) u⋅v=|u||v|cosθ, where θ is the angle between the vectors. Notice that the dot product is zero if the two vectors are perpendicular to each other, and equals the product of their absolute values if they are parallel.
• 14.3: The Vector Product
The vector product of two vectors is a vector defined as u×v=|u||v|n sin θ, where θ is again the angle between the two vectors, and n is the unit vector perpendicular to the plane formed by u and v. The direction of the vector n is given by the right-hand rule.
• 14.4: Vector Normalization
A vector of any given length can be divided by its modulus to create a unit vector (i.e. a vector of unit length).
• 14.5: Problems

This page titled 14: Vectors is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Marcia Levitus via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.