Vector-Valued Functions
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Transcript Vector-Valued Functions
10.3 Vector Valued Functions
Greg Kelly, Hanford High School, Richland, Washington
Any vector
v a, b
can be written as a linear
combination of two standard unit vectors.
i 1,0
v a, b
a,0 0, b
a 1,0 b 0,1
ai bj
j 0,1
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.
v a, b
a,0 0, b
Either of these is an acceptable
way to express the same vector
function.
a 1,0 b 0,1
ai bj
We can describe the position of a moving particle by a
vector, r(t).
r t
f t i
g t j
r t f t i g t j
or
r(t) = f (t), g (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.
In three dimensions the component form becomes:
r t f t i g t j h t k
Given the position vector:
r(t) = f (t), g (t)
which we can also write as… r(t) = x(t), y(t)
The velocity vector would be:
The acceleration vector would be:
dx dy
v(t) =
,
dt dt
d 2x d 2 y
, 2
a(t) =
2
dt dt
Graph on the TI-83 using the parametric mode.
r t t cos t i t sin t j
t 0
Use this
window setting:
This is just 8p
Graph on the TI-89 using the parametric mode.
r t t cos t i t sin t j
t 0
Hitting zoom fit followed by zoom square will give us…
Most of the rules for the calculus of vectors are the same as
we have used, except:
Speed v t
“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.
v t
velocity vector
Direction
v t
speed
Most of the rules for the calculus of vectors are the same as
we have used, except:
Speed v t
dx dy
v(t) =
,
dt dt
Since we know what the
components of v(t) are…
2
dx dy
Speed v t
dt dt
2
v t
velocity vector
Direction
v t
speed
r t 2t 3 3t 2 i t 3 12t j
a) Write the equation of the tangent where
t 1.
dr
v t
6t 2 6t i 3t 2 12 j
dt
At t 1 :
position:
tangent:
v 1 12i 9j
r 1 5i 11j
5,11
slope:
y y1 m x x1
3
y 11 x 5
4
9
3
12
4
3
29
y x
4
4
r t 2t 3 3t 2 i t 3 12t j
dr
v t
6t 2 6t i 3t 2 12 j
dt
b) Find the coordinates of each point on the path where
the horizontal component of the velocity is 0.
2
6
t
6t .
The horizontal component of the velocity is
6t 6t 0
2
t t 0
r 0 0i 0j
0, 0
2
t t 1 0
t 0, 1
r 1 2 3 i 1 12 j
r 1 1i 11j
1, 11
dr
v t
6t 2 6t i 3t 2 12 j
dt
The velocity vector is often called the tangent vector.
Now let’s try an initial value problem:
1
v =
i 2t j
t 1
r(0) = i – 2j
Find the vector function r (t)
dt
r =
i
t 1
2t dt j
ln(t 1) C1 i t 2 C2 j
r(0) = i – 2j ln(0 1) C1 i 02 C2 j
C1 1
C2 2
r ln(t 1) 1 i t 2 2 j
p