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How do you integrate a vector-valued function \(r(t) = \langle f(t), g(t), h(t) \rangle\)?
\(\int r(t) dt = \langle \int f(t) dt, \int g(t) dt, \int h(t) dt \rangle\)
What is the distance formula in 2D?
\(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
How do you find the position of a particle at time t given its velocity vector and initial position?
1. Integrate the velocity vector to find the position vector. 2. Use the initial position to solve for the constant of integration. 3. Plug in t to find the position at that time.
How do you find the distance of a particle from the origin at time t, given its velocity vector and initial position?
1. Integrate the velocity vector to find the position vector. 2. Use the initial position to solve for the constant of integration. 3. Plug in t to find the position at that time. 4. Use the distance formula to find the distance from the origin.
How are the components of a vector-valued function integrated?
Each component is integrated separately, treating each as its own integral problem.
What do the constants of integration represent when integrating a vector-valued function representing velocity?
The initial position of the particle.
What information can be obtained by integrating a vector-valued acceleration function?
The velocity function, plus a constant of integration representing initial velocity.