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Why do 3-lines have various slopes?
Slope is calculated against an axis. Since there are 3, there are many slopes.
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How do you find the angle between vectors v and i?
cos  = v 1/||v||
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How do you find the angle between vectors v and j?
cos = v 2/||v||
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How do you find the angle between vectors v and k?
cos = v 3/||v||
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v = <4,3,6>
What are the angles of i, j and k?
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Define line
collection of points in a certain direction
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Find the equation of line through (-1,-2) in direction of <-2,-4>
y = -2x - 4
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What do parametric equations show?
movement through time.
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x = x1 + at
y = y1 + bt
what does x1, y1 and a,b represent? How do you find the slope?
- x1,y1 are points
- a,b are the x and y components
- b/a
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(2,-8) <3,2>
Find the parametric equations.
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P = (3,1)
Q = (5,6)
Find the parametric equations.
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Why does it not matter which point you pick when finding a parametric?
Both points are on the equations(parametric line). Choosing one over the other just changes the time at that point on the line.
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What is the equation for parametric form?
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What is the equation for Symmetric form?
- (x-x1)/a = (y-y1)/b = (z-z1)/c
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(1,4) <2,5>
Find the equation in parametric and symmetric form.
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(-2,1,3)
<4,-2,1>
Find the parametric and symmetric form of the equation.
- x = -2 + 4t
- y = 1 - 2t
- z = 3 + t
(x+2)/4 = (y-1)/-2 = (z-3)
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What is the equation for a plane?
a(x-x1) + b(y-y1) + c(z-z1)
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What is the general equation for a plane?
ax + by + cz +d =0
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Define plane.
Given a point(p) and vector(v). A plane contains all p vectors perpendicular to v.
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P = (1,-2,3)
Q = (0,2,1)
R = (-3,4,1)
Find the equation of the plane.
2x + 3y + 5x - 11
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Find the parametric and symmetric equations through the points.
(4,-3,-2) (-2/3,2/3,1)
x = -14t+4, y=11t-3, z=9t-2
(4-x)/14 = (y+3)/11 = (z+2)/9
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z =5.
what are the intercepts?
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Determine any planes that are parallel or identical.
P1: -60x + 90y + 30z = 23
P2: 4x - 6y - 2z = 9
P3: -20x + 30y + 10z = 7
P4: 12x - 18y + 6z = 3
P1, P2, P3
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Determine any lines that are parallel or identical.
L1: (x − 4)/6 = (y + 7)/−3 = (z + 8)/7
L2: (x + 4)/4 = y − 3 = (z + 5)/9
L3: (x + 7)/-12 = (y − 50/6 = (z + 16)/-14
L4: (x − 4)/-6 = (y + 6)/1 = (z − 2)/3.5
L1 and L3
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Find sets of parametric equations and symmetric equations of the line
through the point parallel to the given vector or line.
Point (0,0,0) Parallel to <4,1,2>
x=4t , y=t , z=2t
x/4 = y = z/2
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Find a set of parametric equations of the line that passes through the point (−1, 4, -9) and is parallel to v = 6i − j.
x = 6t-1, y=-t+4, z=-9
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Find an equation of the plane passing through the point perpendicular to the given vector or line.
Point (0, 6, 0) Perpendicular to n = −5i + 3k
0 = 3z - 5x
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Find an equation of the plane that passes through the point (3, 6, 7) and is parallel to the yz-plane.
x=3
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How do you tell if 2 symmetric equations are equal?
They're equal if their dot product equals 0.
(dot product of the denominator)
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Find an equation of the plane that passes through the point (6, -4, 3) and contains the line given by the following equation.
x/2 = (y-4)/-1 = z
0 = -5x + 10z
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Find the equation of the plane passes through the points (5, 3, 1) and (5, 1, -6) and is perpendicular to the plane 8x + 9y + 3z = 17.
57x − 56y + 16z = 133
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How do you sketch the graph for an equation?
set 2 of the variables to zero, then solve for the third. Repeat to find other 2 variables.
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How do you sketch the graph for a symmetrical equation?
Set one variable to 0, then solve for t. Plug that t into the other 2 equations to solve for those values. Those are the coordinates in the dimension. Repeat for the other 2.
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Find an equation of the plane passing through the point perpendicular to the given vector or line.
(8, 7, 7)
Perpendicular to: (x − 1)/14 = y + 7 = (z + 8)/-8
14x + y − 8z = 63
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Find the coordinates of a point P on the line and a vector v parallel to the line.
(x − 5)/6 = (y + 6)/7 = z + 1
- P = (5, −6, −1) (other answers possible)
- v = 6, 7, 1 (any nonzero multiple of v is correct)
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