Chapter 13

Question 1

Unit vector

Answer 1

vector with length equal to 1

Question 2

|u| =

Answer 2

sqr root of (u1^2 + u2^2 + u^3), magnitude/length/norm

Question 3

what scalar multiple of v gives a unit vector in the same direction as v?

Answer 3

(1/(|v|))(|v|), v != 0

Question 4

dot product of u and v

Answer 4

u1(v1) + u2(v2) + u3(v3) -> gives a scalar (number)

Question 5

if two vectors, u and v, are orthogonal

Answer 5

then the dot product of u and v = 0

Question 6

projvU (vector projection of u onto v) =

Answer 6

scalvU(v/|v|)

Question 7

the work done by a constant force F moving an object along the displacement vector D =

Answer 7

(the amount of force in d direction) (|d|) = |F||D|cos theta = F*d

Question 8

cross product of u and v, u x v is

Answer 8

the vector with magnitude |u x v| = |u|*|v| sin(theta) –> scalar as answer

Question 9

cross product given u = <u1, u2, u3> and v = <v1, v2, v3>

Answer 9

u x v = <u3v2 - u2v3, u3v1 - u1v3, u1v2 - u2v1> –> vector as answer

Question 10

volume =

Answer 10

base area * height = |(u x v)| * |w||cos theta| =

Question 11

In R^3, determine a line by

Answer 11

A) a point on it and B) direction via vector parallel to the line

Question 12

In R^3, any two ____ determine _____

Answer 12

points, line

Question 13

In R^3, any three ____ determine _____

Answer 13

non-collinear points, plane

Question 14

In R^3, determine an equation for plane from

Answer 14

A) a point on it and B) a vector normal/orthogonal to it

Question 15

Let (x0, y0, z0) be a point on line l and vector v = <a, b, c> be in the directional of l. An arbitrary point on l must satisfy

Answer 15

vector P = vector P0 + t(vector v)
<x, y, z> = <x0+ ta, y0 + tb, z0 + tc> –> vector equation
x = x0 + a y = y0 + b z = z0 + z –> parametric equations
eliminate parameter: t = (x - x0)/a = (y-y0)/b = (z-z0)/c –> symmetric equations

Question 16

Let (x0, y0, z0) be a point on a plane and n = <a, b, c> be a vector that is orthogonal (a normal vector) to the plane. An arbitrary point (x, y, z) on the plane must satisfy

Answer 16

P0P * n = 0,
<x-x0, y-y0, z-z0> * <a, b, c> = 0
= a(x - x0) + b(y - y0) + c(z - z0) = 0
ax + by + cz = d, where d = ax0 + by0 + cz0

Question 17

Give an equation for the plane containing points: P, Q, and R.

Answer 17

vector PQ, vector QR, vector PR would be in the plane, so the cross product of any 2 is normal to the plane
1) Find the vectors PQ, QR, PR (any two of them)
ex) vector PQ = <q1-p1, q2-p2, q3-p3>
vector RQ = <q1-r1, q2-r2, q3-r3>
2) Find the cross product of those vectors
ex) vector PQ x vector RQ:
<(q2-p2) * (q3-p3) - (q3-p3) * (q2-r2) i,
(q1-p1) * (q3-r3) - (q2-p2) * (q1-r1) j,
(q1-p1) * (q2-r2) - (q2-p2) * (q1-r1) k >
Simplifies to vector v = <#, #, #>
3) Plug into formula for equation:
a(x - x0) + b(y - y0) + c(z - z0) = ax + by + zc =
constant

Question 18

Normal/orthogonal =

Answer 18

perpendicular to the tangent line to the curve at the point

Question 19

Vector valued function is

Answer 19

a function of a real variable that results in a vector
Notation = vector r(t) = <x(t), y(t), z(t)>, x y and z are real valued functions of a single variable

Question 20

ellipsoid

Answer 20

((x - h)^2/ a^2) + ((y - k)^2/b^2) + ((z - l)^2/c^2) = 1
center: (h, k, l)

Question 21

cone

Answer 21

((x - h)^2/ a^2) + ((y - k)^2/b^2) = ((z - l)^2/c^2)
center: (h, k, l)

Question 22

cylinder

Answer 22

((x - h)^2/ a^2) + ((y - k)^2/b^2) = 1
center: (h, k)

Question 23

elliptical paraboloid

Answer 23

((x - h)^2/ a^2) + ((y - k)^2/b^2) = ((z - l)/c)
center: (h, k, l)

Question 24

hyperbolic paraboloid

Answer 24

((x - h)^2/ a^2) - ((y - k)^2/b^2) = ((z - l)/c)
center: (h, k, l)

Question 25

unit vector =

Answer 25

vector u/|u|

Question 26

vector S =

Answer 26

vector b (speed) + vector w (water)

Question 27

angle between 2 vectors =

Answer 27

cos theta = |uv|/(|u||v|), theta = arccos(|uv|/(|u||v|))

Question 28

radius of sphere given center and pt

Answer 28

r = square root ( (x1-x2)^2+(y1-y2)^2+(z1-z2)^2 )

Question 29

unit tangent vector =

Answer 29

derivative of r(t)/|derivative of r(t)

Question 30

definite integral that gives length of curve given r(t)

Answer 30

integral from a to b (|derivative of r(t)| dt

Question 31

if r(t) is parametrized by arc length

Answer 31

then |derivative of r(t)| = 1