Question

Find the point P on the curve r(t) that lies closest to P and state the distance between P₀ and P.

r(t) = t² i+t j + t k, P₀(1,12,4)

Answer 1

To find the point P on the curve r(t) that lies closest to P₀, we can use the formula for the projection of a point onto a line. We differentiate the equation for the distance between P and P₀ with respect to t and set the derivative equal to zero. After solving the equation, we can find the value of t that minimizes the distance and then find the coordinates of the point P.

Step-by-step explanation:

To find the point P on the curve r(t) that lies closest to P₀, we need to minimize the distance between P and P₀. This can be done by finding the shortest distance between P and P₀, which is along the line segment connecting P and P₀. The point P is the projection of P₀ onto the curve.

To find P, we can use the formula for the projection of a point onto a line. Let P(t) be the point on the curve at time t. The distance between P and P₀ is given by the equation d(t) = ||P(t) - P₀||, where ||v|| represents the magnitude of vector v. We want to minimize d(t), so we can differentiate it with respect to t and set the derivative equal to zero.

After solving the equation, we can find the value of t that minimizes d(t). Substituting this value of t into the equation for P(t), we can find the coordinates of the point P.