Vector Basics
Explain the difference between a vector and a scalar.
Vector: magnitude (length) and direction
Scalar: just a magnitude / number, no direction
Find the dot product between <3,2,-1> and <-4,-2,0>.
-16
In R2, a set of parametric equations can be thought of as having how many inputs and how many outputs?
One input: t (time)
Two outputs: (x,y) (position)
Determine r(pi) for:
r(t) = <sin(2t), cos(2t), 3>.
r(pi) = <0, 1, 3>
Find the length of: <-4,5,-1>
sqrt(42)
Describe conceptually what the scalar value of a dot product between two vectors tells us.
How "closely" two vectors point in the same direction.
Positive: more in same direction (angle less than pi/2)
0: vectors are orthogonal
Negative: more in opposite directions (angle greater than pi/2)
Eliminate the parameter and rewrite as a single equation of x and y in terms of x:
x=3t+1
y=t5
y=((x-1)/3)5
or
x=3*(5th root of y)+1
Given a vector-valued position function r(t), describe how to find the velocity, speed, and acceleration.
Velocity: first derivative r'(t)
Speed: magnitude of velocity |r'(t)|
Acceleration: second derivative r''(t)
Give two examples of vectors parallel to <3,-2,0>.
any two scalar multiples of <3,-2,0> will work
Compute the cross product:
<1, 2, 3> x <3, 4, 5>
<-2, 4, -2>
Give a set of parametric equations with range on t for the upper half of a circle centered at (1,0) with radius 4, oriented clockwise.
one possibility:
x = 1+ 4sin(t)
y = 4cos(t)
on 0 ≤ t ≤ pi
Differentiate:
r(t) = <ln(1-t), t^0.5, e^3 >
r'(t) = <-1/(1-t), 1/2t^(-1/2), 0>
Find two unit vectors parallel to:
<4, -1, 2>
<4, -1, 2> / sqrt(21)
and
- <4, -1, 2> / sqrt(21)
Determine the numerical value of "a" required so that the cross product between <2, 3, 0 > and <-1, a, 0 > has no length.
a = -3/2
Give a set of parametric equations for the line segment from P(-4,8) to Q(-8,4).
One example:
x = -4 - 4t
y = 8 - 4t
on 0 < t < 1
Find the unit tangent vector to:
r(t) = <te^t, t^2-5t >
at t=0.
<1, -5> / sqrt(26)
Find a vector of length 5 in the opposite direction as the vector pointing from P(3,-1,9) to Q(-3,1,8).
-5<-6,2,-1>/sqrt(41)
or <30,-10,5>/sqrt(41)
Determine a vector orthogonal to <2, 7, -1>
any vector v such that <2, 7, -1> dot v = 0
ex: v = <-3, 1, 1>
Find the arc length of the curve:
r(t) = <3t, -2/3*(7-t)^(3/2) > from t=15 to t=16
2/3
Find the indefinite integral of:
r(t) = < sin^2(t), (t+1)^(3/2), 4/t >
< 1/2*t-1/4*sin(2t), 2/5*(t+1)^(5/2), 4ln|t| >
+ <C1, C2, C3>