Study your flashcards anywhere!
Download the official Cram app for free >
 Shuffle
Toggle OnToggle Off
 Alphabetize
Toggle OnToggle Off
 Front First
Toggle OnToggle Off
 Both Sides
Toggle OnToggle Off
 Read
Toggle OnToggle Off
How to study your flashcards.
Right/Left arrow keys: Navigate between flashcards.right arrow keyleft arrow key
Up/Down arrow keys: Flip the card between the front and back.down keyup key
H key: Show hint (3rd side).h key
A key: Read text to speech.a key
37 Cards in this Set
 Front
 Back
 3rd side (hint)
DISTANCE FORMULA p1, p2

√(x2  x1)² + (y2  y1)² + (z2  z1)²

2 minus 1


Equation of a Sphere with center (h,k,l) and radius r

(x  h)² + (y  k)² + (z  l)² = r²



vector

a quantity that has both magnitude and direction



vector components

<a1, a2, a3>



Definition of vector addition

If a and v are vectors positioned so the initial point of v is at the terminal point of a then the sum u + v is the vector from the initial point of u to the terminal point of v.
PQ + QR = PR QS  PS = QP 


Properties of vectors

a + b = b + a
a + (b + c) = (a + b) + c a + 0 = a a + a = 0 scalarc(a + b) = ca + cb scalars(c + d)a = ca + da scalars(cd)a = c(da) 1a = a 


standard basis vectors

i <1,0,0>
j <0,1,0> k <0,0,1> 


unit vectors

a vector whose length is 1
(standard basis vectors) 


resultant force

the vector sum of several forces acting on an object



position vector

the representation of a vector from the origin.
Given 2 points a = <x2  x1, y2  y1, z2  z1> 


Magnitude or length of a vector

a = √a1² + a2² + a3²



Adding vectors algebraically

a + b = <a1 + b1, a2 + b2, a3 + b3>



dot product

a ∙ b = (a1)(b1) + (a2)(b2) + (a3)(b3)



Properties of the dot product

a ∙ a = a²
a ∙ b = b ∙ a a ∙ (b + c) = a ∙ b + a ∙ c (ca) ∙ b = c(a ∙ b) = a ∙ (cb) 0 ∙ a = 0 


Theorem 3 If Θ is the angle between vectors a and b

a ∙ b = a b cos Θ
cos^1 Θ = a ∙ b ∕ a b 


Scalar projection of b onto a

comp(a) b = a ∙ b / a



vector projection of b onto a

proj a b = (a ∙ b / a²) ( a)



Work/Force/Distance Equation

W = F ∙ D if no angle
or w = FD cos Θ 


Theorem 4 If Θ is the angle between a and b then

a x b = a b sin Θ
or sin^1 Θ = a x b / a b 


x²/a² + y²/b² + z²/c² = 1

Ellipsoid

(3)2nd order terms and a constant)


z²/c² = x²/a² + y²/b²

Cone

(3)2nd order variables
No constants. If put in form x²/a² + y²/b²  z²/b² = 0  Opens towards axis of negative variable. 

z/c = x²/a² + y²/b²

Elliptic paraboloid

(2)2nd order terms that have same sign.
(1)1st order term. No constants. Opens along 1st order term axis. 2nd order terms determines + or . 

z/c = x²/a²  y²/b²

Hyperbolic Paraboloid

(2)2nd order terms that have opposite signs.
(1)1st order term. No constants. Neg 2nd order term = head of horse. 1st order term = top of saddle. 

x²/a² + y²/b²  z²/c² = 1

Hyperboloid of one sheet

(3)2nd order terms
(1) constant (1) negative term  opens along this term 

x²/a²  y²/b² + z²/c² = 1

Hyperboloid of two sheets

(3) 2nd order terms
(1) constant (1) positive term  opens along this term 

x² + y² = 1

Cylinder

(2) variables
extruded along missing variable 

vector equation of a line

r = rօ + t(v)
r = (x,y,z) + t<a,b,c> 


parametric equations

x = xօ + at
y = yօ + bt z = zօ + ct 


symmetric equations

(x  xօ)/a = (y  yօ)/b = (z  zօ)/c



vector equation of a plane

n ∙ r = n ∙ rօ
or n ∙(r  rօ) = 0 


scalar equation of a plane

a(x  xօ) + b(y  yօ) + c(z  zօ) = 0



normal vector

vector orthogonal to a plane

cross product


linear equation of a plane

ax + by + cz + d = 0



parallel planes

normal vectors are parallel



Distance from point to a plane

ax₁ + by₁ + cz₁ + d/√a² + b² + c²



Area of a parallelogram

a x b



volume of a parallelepiped

a ∙ (b x c)

