Glossary
Acceleration Vector/Function
A vector-valued function, a(t), that describes the instantaneous rate of change of a particle's velocity, indicating how its velocity is changing.
Example:
If a particle's velocity is v(t) = ⟨t², 3t⟩, its acceleration vector a(t) would be ⟨2t, 3⟩, found by differentiating v(t).
Component-wise Integration
The method of integrating a vector-valued function by integrating each of its scalar components independently.
Example:
When integrating v(t) = ⟨3t², 3⟩, you apply component-wise integration by integrating 3t² with respect to t for the x-component and 3 with respect to t for the y-component.
Constant of Integration
An arbitrary constant (or constants, C1, C2, etc.) that arises from indefinite integration, representing the family of all possible antiderivatives.
Example:
After integrating v(t) = ⟨3t², 3⟩ to get r(t) = ⟨t³ + Cₓ, 3t + Cᵧ⟩, Cₓ and Cᵧ are the constants of integration that define the specific path.
Distance Formula (in context of vectors)
Used to calculate the straight-line distance between two points in a coordinate plane, often applied to find the distance of a particle from the origin or another point.
Example:
To find how far a particle at (9,8) is from the origin (0,0), you would use the distance formula: √((9-0)² + (8-0)²).
Initial Condition
A specific value of a function or its derivative at a given point, used to determine the unique constant(s) of integration for a particular solution.
Example:
If a particle is at point (1,2) at t = 0, this initial condition allows you to solve for the specific constants of integration in its position vector.
Integrating Vector-Valued Functions
The process of finding the antiderivative of a vector-valued function, often used to determine position from velocity or velocity from acceleration.
Example:
To find a particle's velocity from its acceleration a(t) = ⟨2t, 5⟩, you would perform integrating vector-valued functions on a(t).
Position Vector/Function
A vector-valued function, r(t), that describes the location of a particle or object in space at any given time t.
Example:
If r(t) = ⟨t³ + 1, 3t + 2⟩, then r(t) is the position vector of a particle, telling you exactly where it is at time t.
Vector-Valued Function
A function whose output is a vector, typically dependent on a single scalar variable (often time, t). It describes motion or position in 2D or 3D space.
Example:
If a particle's path is given by r(t) = ⟨t², sin(t)⟩, then r(t) is a vector-valued function describing its position at any time t.
Velocity Vector/Function
A vector-valued function, v(t), that describes the instantaneous rate of change of a particle's position, indicating both its speed and direction.
Example:
Given a particle's position r(t) = ⟨sin(t), cos(t)⟩, its velocity vector v(t) would be ⟨cos(t), -sin(t)⟩, found by differentiating r(t).