Vector products are crucial in physics, allowing us to describe complex interactions between forces and objects. They come in two flavors: scalar products (dot products) and vector products (cross products), each with unique properties and applications.
These mathematical tools help us calculate work, , and . Understanding vector products is key to grasping mechanics, as they provide a powerful way to analyze forces and motion in three-dimensional space.
Vector Products
Scalar vs vector products
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2.4 Products of Vectors | University Physics Volume 1 View original
() yields a scalar quantity denoted by A⋅B and calculated as ∣A∣∣B∣cosθ where θ is the angle between vectors
Commutative property: A⋅B=B⋅A
Determines work done by a force W=F⋅d and of one vector onto another
() yields a vector quantity denoted by A×B with ∣A∣∣B∣sinθ
perpendicular to the plane containing A and B determined by the
Anti-commutative property: A×B=−(B×A)
Calculates torque τ=r×F, L=r×p, and magnetic force on a moving charge F=qv×B
Vector algebra
of a vector represents its length or size
Direction of a vector indicates the orientation in space
is used for vector addition
applies: A⋅(B+C)=A⋅B+A⋅C
Calculation of scalar products
Calculate using vector components: A⋅B=AxBx+AyBy+AzBz
Calculate using magnitudes and angle between vectors: A⋅B=∣A∣∣B∣cosθ
Positive indicates force has a component in the same direction as displacement (work done)
Negative scalar product indicates force has a component opposite to displacement direction (work against)
Zero scalar product means force is perpendicular to displacement resulting in no work done
Determines the component of A along the direction of B through projection ∣A∣cosθ
Computation of vector products
Calculate using vector components: A×B=(AyBz−AzBy)i^−(AxBz−AzBx)j^+(AxBy−AyBx)k^
Calculate using the determinant of a 3x3 matrix:
A×B=i^AxBxj^AyByk^AzBz
Magnitude ∣A×B∣=∣A∣∣B∣sinθ represents the area of the parallelogram formed by A and B
Direction perpendicular to the plane containing A and B determined by the right-hand rule (point fingers along A, curl towards B, thumb points in direction of A×B)
Vector products in mechanics
Torque calculated as τ=r×F where r is position vector from axis of rotation to force application point and F is the applied force
Magnitude ∣τ∣=∣r∣∣F∣sinθ where θ is angle between r and F
Direction perpendicular to plane containing r and F determined by right-hand rule
Angular momentum calculated as L=r×p where r is position vector from origin to particle and p is linear momentum of particle
Magnitude ∣L∣=∣r∣∣p∣sinθ where θ is angle between r and p
Direction perpendicular to plane containing r and p determined by right-hand rule
Conservation of angular momentum: total angular momentum of a system remains constant in the absence of external torques