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distance between 2 ordered triplets
d=√( (X2-X1)2 + (Y2-Y1)2 + (Z2-Z1)2 )
equation of sphere centered at (-5,6,-8) with rad 7
(x+5)2 + (y-6)2 + (z+8)2 = 49
if equation of object = x+5)2 + (y-6)2 + (z+8)2 = 49, how do you find the shape shown in the y-z plane?
set x=0
standard equation of an Ellipsoid
(X2/A2) + (Y2/B2) + (Z2/C2) = 1
standard equation of an Hyperboloids of 1 sheet
(X2/A2) + (Y2/B2) - (Z2/C2) = 1
hour glass graph that doesn\'t separates in the center
Hyperboloid of 1 sheet
standard equation of an Hyperboloids of 2 sheet
-(X2/A2) - (Y2/B2) + (Z2/C2) = 1
hour glass graph that separates in the center
Hyperboloid of 2 sheet
standard equation of an Cone
(X2/A2) + (Y2/B2) = Z2
standard equation of an Elliptic Paraboloid
(X2/A2) + (Y2/B2) = (Z/C)
bowl graph
Elliptic Paraboloid
standard equation of an Hyperbolic Paraboloid
(X2/A2) - (Y2/B2) = Z/C
Potato chip graph
Hyperbolic Paraboloid
Find the unit vector that has the same direction as u. u=−2,−5,7
find magnitude and divide vector by it … 1/√78 <-2,-5,7>
do 2.3
when you know vectors U and V, and you want to find the direction of U+V use
u1v1 + u2v2 + u3v3
formula for the angle between two vectors
cosø = (|n1 DOT n2|) / (||n1|| ||n2||)
U dot V alt fourmula
= ||u|| ||v|| cosø
The vectors u and v are orthogonal if u • v =
uX0 =
uXu =
uXv = (in terms of v & u)
||uXv|| =
||u|| ||v|| sinø
v( u dot v / ||v||2)
u dot v / ||v||
Work (3 formulas!)
||PROJpqF|| Times ||PQ|| ; ||F|| time ||PQ|| times Cosø ; F dot PQ
u×v is orthogonal to
both u and v
U X V =
||u X v|| =
||u|| ||v|| sinø
u×v = 0 if and only if
u and v are scalar multiples of each other
|| u×v|| =
area of parallelogram having u and v as adjacent sides
A line L parallel to the vector v = a,b,c and passing through the point P(x1,y1,z1) is represented by the parametric equation
x = x1 + at y = y1 + bt z = z1 + ct
Suppose a plane contains the point ( x1 , y1 , z1 ) and has a normal vector v = a, b, c Then an equation of the plane is given by
a ( x − x1 ) + b ( y − y1 ) + c ( z − z1 ) = 0
The general form or linear form of the equation of a plane is given by
ax+by+cz+d =0
to find the angle between two intersecting planes by finding
the angle between the normal vectors
Two planes are perpendicular if n1 • n2
= 0 .
Two planes are parallel if n 1 is
a scalar multiple of n 2
The distance between a plane and a point Q (not on the plane) is
||PROJnPQ|| = |PQ dot n| / ||n|| Where P is a point in the plane and n is normal to the plane.
The distance between a point Q and line in space is given by
||PQ X V|| / ||v|| Where v is the direction vector for the line and P is a point on the line.
do a problem from 12.6
Suppose r(t) represents a smooth curve on an open interval. The unit tangent vector is given by
T(t) = r’(t) / ||r’(t)||
Let r(t) define a smooth curve in space that is traced exactly once as t increase from t = a to t = b. Then the length of the smooth curve is given by
Let r(t) define a smooth curve in space that is traced exactly once as t increase from t = a to t = b. The arc length function is given by
Let r(s) define a smooth curve where s is the arc length parameter. The curvature at s is given by K =
circle has constant curvature K =
The sharper the curve, the ______ the value of K
The curvature of the curve given by the vector function r(t) is
K = ||r’(t) Cross r’’(t)|| / ||r’(t)||3
Consider the curve given in rectangular coordinates by y = f (x). The curvature at the point (x, y) is given by
K = |y’’| / [1 + (y’)2]^(3/2)
do a problem 13.1
x of y cards Next >|