INTRODUCTION

Question 1

distance formula

Answer 1

d = sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1)^2)

Question 2

equation of a sphere

Answer 2

d = r
center = (h, j, k), @ surface = (x, y, z)
(x-h)^2+(y-j)^2+(z-k)^2 = r^2

Question 3

magnitude of v = <dx, dy, dz>

Answer 3

|v| = sqrt(dx^2+dy^2+dz^2)

Question 4

unit vector

Answer 4

magnitude = 1
getting there: v/|v|

Question 5

standard basis vectors in R3

Answer 5

i = <1, 0, 0>
j = <0, 1, 0>
k = <0, 0, 1>
ie. v = <4, 5, 6> = 4i + 5j + 6k

Question 6

dot product: a*b
a = <xa, ya, za>
b = <xb, yb, zb>

Answer 6

a * b = xaxb + yayb + zazb

Question 7

dot product properties (5)

Answer 7

1. a = <a1, a2, a3>
   a * a = a1^2 + a2^2 + a3^2 = |a|^2
2. a * b = b * a
3. a * (b + c) = a * b + a * c
4. s = scalar
   (sa) * b = s(a * b) = a * (sb)
5. 0 * a = 0

Question 8

cosine theorem
a * b =

Answer 8

|a| * |b| * cos(theta)

Question 9

a and b are orthogonal if

Answer 9

a * b = 0

Question 10

a and b are parallel if

Answer 10

same direction:
theta = 0 so cos(theta) = 1

opposite direction:
theta = pi so cos(theta) = -1

Question 11

radians to degrees

Answer 11

*180/pi

Question 12

scalar projection component of b along a

Answer 12

length or projection
projection = x = |b| cos(theta)
x = (a * b)/|a|

compa(b) = (a * b)/|a|

Question 13

vector projection of b onto a

Answer 13

vector along a with length of projection
proja(b) = compa(b) * (a/|a|)

Question 14

projection with an obtuse angle

Answer 14

b will project in the direction of -a to form its right triangle

compa(b) = will be negative as cos<0
multiply proja(b) by -1

Question 15

(for cross product)
determinant of a 2 x 2 matrix
M = (a, b)
(c, d)

Answer 15

det(M) = ad - bc

|a b| = ad - bc
|c d|

Question 16

cross product

Answer 16

a x b = i(a2b3 - a3b2) - j(a1b3 - a3b1) + k(a1b2 - a2b1)

Question 17

cross product is _____ to both vectors

Answer 17

orthogonal,
a x b = -(b x a)

Question 18

to find if a vector crosses front or back

Answer 18

follow the curve of your right hand

Question 19

sin theorem

Answer 19

|a x b| = |a| * |b| * sin(theta)

Question 20

area of a parallelogram w/ sides a and b

Answer 20

A = |a x b|

Atriangle = “ “ /2

Question 21

volume of a parallelopiped w/ sides a and b

Answer 21

V = a * (b x c)

Question 22

a line is determined by…

Answer 22

a point on the line

a direction vector(parallel and scalar multiple of v by t when v lies on the line)

Question 23

parametric equations for the line

through P0(x0, y0, z0) and any P

parallel to <a, b, c> scalar multiple by t

Answer 23

P0P = <x - x0, y - y0, z - z0>

x - x0 = at, y - y0 = bt, z - z0 = ct

x = x0 + at
y = y0 + bt
z = z0 + ct

Question 24

a plane is determined by…

Answer 24

a point on the plane

a normal vector(perpendicular and scalar multiple of v by t when v lies on the plane)

Question 25

equation of a plane

through P0 = (x0, y0, z0) and any P

w/ normal vector n = <a, b, c> scalar multiple of v by t

Answer 25

P0P = <x - x0, y - y0, z - z0>
(orthogonal to n)

a(x - x0) + b(y - y0) + c(z - z0) = 0

Question 26

angle between planes =

Answer 26

= angle between normal vectors