A tangent vector is a vector that represents the direction and magnitude of the instantaneous rate of change of a curve or surface at a specific point. It is perpendicular to the normal vector and lies in the same plane as the curve or surface at that point.
This is a parametric equation that describes a curve in polar coordinates. The variable r represents the distance from the origin and theta represents the angle from the positive x-axis. The specific values of sin(t) and t/3 will determine the shape and orientation of the curve.
To find the tangent vector, we can take the derivatives of r and theta with respect to t. This will give us the components of the tangent vector in terms of t. We can then plug in a specific value of t to get the actual vector at that point on the curve.
The normal vector is a vector that is perpendicular to the tangent vector at a specific point on a curve or surface. To find the normal vector, we can take the cross product of the tangent vector and the vector representing the direction of increasing theta, which is (cos(t), sin(t)).
The tangent vector has many applications in fields such as physics, engineering, and computer graphics. It can be used to calculate the velocity and acceleration of an object moving along a curved path, determine the direction of a force acting on an object, and to create realistic 3D models of curved surfaces.