4.9 Vector-valued functions

The concept of position vectors is central to understanding movement in the plane or space, such as curves in the plane or space curves. A position vector is a tool that we can use to describe the position of a particle in a plane at any given time. It begins at the origin and ends at the point in space that the particle occupies.

To represent the position of a particle moving in a plane, we can use a vector-valued function: p(t)=x(t)\hat{i} + y(t)\hat{j} or p(t)= \left\langle x(t),\,y(t) \right\rangle.

Here, x(t) and y(t) are parametric functions that describe the x and y coordinates of the particle at any given time t, and i and j are the unit vectors along the x and y axes, respectively.

The magnitude of the position vector at any time t gives us the distance of the particle from the origin. This distance can be calculated if the x and y components of the position vector are known with the Pythagorean Theorem: ||p(t)|| = ||x(t),\,y(t)|| = \sqrt{x^{2}+y^{2}}.

For example:

Suppose that a particle moves in such a way that its x and y coordinates at any time t are given by the functions x(t) = 2t + 1 and y(t) = 3t - 2, respectively. Then, the position vector of the particle at any time t is given by p(t)=(2t + 1)\hat{i} + (3t - 2)\hat{j}.

To find the position of the particle at t = 2, we substitute t = 2 into the position vector to get \\p(2) = (2(2) + 1)\hat{i} + (3(2)-2)\hat{j} = 5\hat{i} + 4\hat{j} or p(2) = \left\langle 5,\,4 \right\rangle.

This means that at time t = 2, the particle is 5 units along the x-axis and 4 units along the y-axis from the origin.

Examples

Just as the position of a particle can be described by a vector-valued function, so can its velocity. A velocity vector provides information about both the speed and direction of a particle at a given time.

The velocity of a particle moving in a plane can be expressed by the vector-valued function v(t)=\left\langle x(t),\,y(t) \right\rangle.

The direction of the velocity vector at a certain time indicates the direction of the particle's motion at that time. The sign of x(t) tells us whether the particle is moving to the right (positive) or to the left (negative), and the sign of y(t) tells us whether the particle is moving upwards (positive) or downwards (negative).

The magnitude of the velocity vector at a given time provides the speed of the particle at that time. This can be computed by finding the square root of the sum of the squares of the x and y components of the velocity vector.

Examples