Summer Calculus II -- Vector / Parametric Equations Review Extravaganza!!

Vector Basics

100

Explain the difference between a vector and a scalar.

Vector: magnitude (length) and direction

Scalar: just a magnitude / number, no direction

200

Find the length of: <-4,5,-1>

sqrt(42)

300

Give two examples of vectors parallel to <3,-2,0>.

any two scalar multiples of <3,-2,0> will work

400

Find two unit vectors parallel to:

<4, -1, 2>

<4, -1, 2> / sqrt(21)

and

- <4, -1, 2> / sqrt(21)

500

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)

Dot / Cross Products

100

Find the dot product between <3,2,-1> and <-4,-2,0>.

-16

200

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)

300

Compute the cross product:

<1, 2, 3> x <3, 4, 5>

<-2, 4, -2>

400

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

500

Determine a vector orthogonal to <2, 7, -1>

any vector v such that <2, 7, -1> dot v = 0

ex: v = <-3, 1, 1>

Parametric Curves

100

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)

200

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

300

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

400

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

500

Find the arc length of the curve:

r(t) = <3t, -2/3*(7-t)^(3/2) > from t=15 to t=16

2/3

Vector-Valued Functions

100

Determine r(pi) for:

r(t) = <sin(2t), cos(2t), 3>.

r(pi) = <0, 1, 3>

200

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)

300

Differentiate:

r(t) = <ln(1-t), t^0.5, e^3 >

r'(t) = <-1/(1-t), 1/2t^(-1/2), 0>

400

Find the unit tangent vector to:

r(t) = <te^t, t^2-5t >

at t=0.

<1, -5> / sqrt(26)

500

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>