11.5

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 Author: NateGatsby ID: 196585 Filename: 11.5 Updated: 2013-02-16 23:44:45 Tags: calc1 Folders: Description: calc1 Show Answers:

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1. Why do 3-lines have various slopes?
Slope is calculated against an axis. Since there are 3, there are many slopes.
2. How do you find the angle between vectors v and i?
cos = v1/||v||
3. How do you find the angle between vectors v and j?
cos = v2/||v||
4. How do you find the angle between vectors v and k?
cos = v3/||v||
5. v = <4,3,6>

What are the angles of i, j and k?
• 59.2 degrees
• 67.4
• 39.8
6. Define line
collection of points in a certain direction
7. Find the equation of line through (-1,-2) in direction of <-2,-4>
y = -2x - 4
8. What do parametric equations show?
movement through time.
9. 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
10. (2,-8) <3,2>

Find the parametric equations.
• x = 2 + 3t
• y = -8 + 2t
11. P = (3,1)
Q = (5,6)

Find the parametric equations.
• x = 3 + 2t
• y = 1 + 5t
12. 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.
13. What is the equation for parametric form?
• x = x1 + at
• y = y1 + bt
14. What is the equation for Symmetric form?
• (x-x1)/a = (y-y1)/b = (z-z1)/c
15. (1,4) <2,5>

Find the equation in parametric and symmetric form.
(x-1)/2 = (y-4)/5

• x = 1 + 2t
• y = 4 + 5t
16. (-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)
17. What is the equation for a plane?
a(x-x1) + b(y-y1) + c(z-z1)
18. What is the general equation for a plane?
ax + by + cz +d =0
19. Define plane.
Given a point(p) and vector(v). A plane contains all p vectors perpendicular to v.
20. P =  (1,-2,3)
Q = (0,2,1)
R = (-3,4,1)

Find the equation of the plane.
2x + 3y + 5x - 11
21. 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
22. z =5.

what are the intercepts?
• x= 0
• y = 0
• z = 5
23. 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
24. 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
25. 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
26. 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
27. 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
28. Find an equation of the plane that passes through the point (3, 6, 7) and is parallel to the yz-plane.
x=3
29. How do you tell if 2 symmetric equations are equal?
They're equal if their dot product equals 0.

(dot product of the denominator)
30. 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
31. 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
32. 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.
33. 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.
34. 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
35. 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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