Vector Functions A vector function is a vector whose components are real-valued functions of a common variable (parameter), usually t.  We’ve seen a vector.

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Presentation transcript:

Vector Functions A vector function is a vector whose components are real-valued functions of a common variable (parameter), usually t.  We’ve seen a vector function before… the parametric equations of a line A general vector function would be

These are vector functions because, when we evaluate r(t) at a value of t, the result is a vector. The domain of r(t) is the intersection of the domains of the component functions.

Ex. Consider, find a) r(2) b)|r(t)| c)the domain of r(t)

Limits behave as we’d expect. Ex. Find where

Ex. Represent the curve as a vector function: a) y = x 2 b) x 2 + y 2 = 25

The graph of a vector function is called a space curve. Ex. Sketch and indicate the orientation of the curve whose vector equation is (first find rectangular)

Ex. Describe the curve whose vector equation is

Ex. Find the vector function for the curve of intersection of the cylinder x 2 + y 2 = 1 and the plane y + z = 2.

Derivatives and integrals behave as we’d like. Ex. Consider, find a) r(t) b)The unit tangent vector at the point where t = 0

Ex. Find the parametric equations of the line tangent to at

Ex. The functions and intersect at the point (1,1,1). Find the angle of intersection.

A space curve given by the vector function r(t) is smooth if r(t) is continuous and r(t) ≠ 0. Ex. Find all values of t where is not smooth.

Look on p. 844 for the properties of vector function derivaties (sum, product, etc. rules)

Ex. Show that if |r(t)| = c, then r(t) is orthogonal to r(t) for all t.

Ex.

Ex. Find the antiderivative of that satisfies

If r(t) = displacement vector, then r(t) = velocity v(t) r  (t) = accelerationa(t) ||r(t)|| = speedv(t)

Ex. Find the velocity, acceleration, and speed of a particle with position vector

Ex. A particle starts with initial position and initial velocity. If its acceleration is, find the position and velocity functions.

Newton’s Second Law of Motion: F(t) = ma(t)

Ex. A projectile is fired with muzzle speed 150 m/s and an angle of 45º from a position 10m above the ground. Where does the projectile hit the ground, and with what speed?