10.3 Vector Valued Functions Greg Kelly, Hanford High School, Richland, Washington

Any vector can be written as a linear combination of two standard unit vectors. The vector v is a linear combination of the vectors i and j. The scalar a is the horizontal component of v and the scalar b is the vertical component of v.

Either of these is an acceptable way to express the same vector function.

We can describe the position of a moving particle by a vector, r ( t ). If we separate r ( t ) into horizontal and vertical components, we can express r ( t ) as a linear combination of standard unit vectors i and j. or r(t) =  f (t), g (t) 

In three dimensions the component form becomes: r(t) =  f (t), g (t)  v(t) = a(t) = Given the position vector: The velocity vector would be: The acceleration vector would be: r(t) =  x(t), y(t)  which we can also write as…

Graph on the TI-83 using the parametric mode. Use this window setting: This is just 8 

Graph on the TI-89 using the parametric mode. Hitting zoom fit followed by zoom square will give us…

And since this also tells us that speed is the ________ of velocity, we must use ____________________. Most of the rules for the calculus of vectors are the same as we have used, except: “Absolute value” means “distance from the origin”. And since this also tells us that speed is the magnitude of velocity, we must use the Pythagorean theorem.

Most of the rules for the calculus of vectors are the same as we have used, except: Since we know what the components of v(t) are… v(t) =

a) Write the equation of the tangent where. At : position: slope: tangent:

The horizontal component of the velocity is. b) Find the coordinates of each point on the path where the horizontal component of the velocity is 0.

The velocity vector is often called the tangent vector.

Now let’s try an initial value problem: v =ij r(0) = i – 2j Find the vector function r (t) r =ij ij r(0) = i – 2jij ijr 