Question

Find a vector parametric equation for the position of the particle (in terms of the original parameter t) as it moves along this tangent line.

Answer 1

To find a vector parametric equation for the position of the particle as it moves along the tangent line, first find the velocity vector by differentiating the position vector with respect to time. Then, find the point on the tangent line and the direction vector of the line. Finally, use the vector parametric equation to describe the position of the particle on the tangent line.

Step-by-step explanation:

To find a vector parametric equation for the position of the particle as it moves along the tangent line, we need to determine the velocity vector first. The velocity vector can be found by taking the derivative of the position vector with respect to time. Given the position vector, ř(t) = (3.0t²î + 5.0ĵ – 6.0tk) m, the velocity vector is obtained by differentiating each component with respect to time:

v(t) = and therefore,

Now, to find the tangent line, we need a point on the line and the direction vector of the line. The point is the position vector at a specific time, and the direction vector is the velocity vector at that same time. Let's say we want to find the tangent line at time t = 0.

Substituting t = 0 into the velocity vector, we get:

v(0) = a₀ = (0î + 5.0ĵ) m/s2

So, the vector parametric equation for the position of the particle as it moves along the tangent line at t = 0 is:

r(t) = r₀ + a₀t = (3.0t²î + (5.0t + 5.0)ĵ – 6.0t)