Given a smooth vector-valued function r(t) , we defined that any vector parallel to r(t₀) is tangent to the graph of r(t) at t=t₀. It is often useful to consider just the direction of r⃗ ′(t) ...Given a smooth vector-valued function r(t) , we defined that any vector parallel to r(t₀) is tangent to the graph of r(t) at t=t₀. It is often useful to consider just the direction of r⃗ ′(t) and not its magnitude. Therefore we are interested in the unit vector in the direction of r(t) . This leads to a definition of the unit tangent vector.
For this topic, we will be learning how to calculate the length of a curve in space. The ideas behind this topic are very similar to calculating arc length for a curve in with x and y components, but ...For this topic, we will be learning how to calculate the length of a curve in space. The ideas behind this topic are very similar to calculating arc length for a curve in with x and y components, but now, we are considering a third component, z.
The derivative of a vector valued function gives a new vector valued function that is tangent to the defined curve. The analog to the slope of the tangent line is the direction of the tangent line. Si...The derivative of a vector valued function gives a new vector valued function that is tangent to the defined curve. The analog to the slope of the tangent line is the direction of the tangent line. Since a vector contains a magnitude and a direction, the velocity vector contains more information than we need. We can strip a vector of its magnitude by dividing by its magnitude.
