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Calc 3 10.5

 

<== Calculus 3
Number Question Answer
See the work
1. Given a the vector equation r(t) = (-3 - 5t)i +
(-2+3t)j+(2t)k,
rewrite this in terms of the parametric equations
for the line.
x(t) =______
y(t) =______
z(t) =______
x(t) = -3 - 5t
y(t) =-2 + 3t
z(t) =2t
The work
2. Given a the vector equation r(t) = (-1 + 2t)i+
(-5 - t)j+(-3+4t)k, rewrite this in terms of the symmetric
equations for the line.
(quotient involving x) = _______
(quotient involving y) = _______
(quotient involving z) = _______

x = (x + 1)/2

y = (y + 5)/-1

z =  (z +3)/4

The work
3. Find the vector and parametric equations for the line through the
point P(-4, 3, -4) and parallel to the vector -4i + 2j + 3k.
Vector Form: r = < _____ , _____ , -4 > + t<_____ ,  _____, 3 >
Parametric form (parameter t, and passing through P when t
= 0):
x = x(t) = _________
y = y(t) = _________
z = z(t) = _________
r = < -4 , 3 , -4 > + t< -4 , 2 , 3 >
x(t) = -4 + -4t
y(t) = 3 + 2t
z(t) =-4 + 3t
The work
4. Find the vector and parametric equations for the line through the
point P(-1, -5, 2) and parallel to the vector < 5, -2, 4 >.
Vector Form: r = < ____, ____, 2 > + t< ____, ____, 4>
Parametric form (parameter t, and passing through P when t
= 0):
x = x(t) = ________
y = y(t) = ________
z = z(t) = ________
r = < -1,  -5, 2 > + t<  5, -2, 4>

x = x(t) = -1 + 5t
y = y(t) = -5 - 2t
z = z(t) = 2 + 4t
The work
5. Find a vector equation for the line through the point
P = (-5, 0, -2) and parallel to the vector v = (5, 5, 2).
Assume r(0) = -5i + 0j - 2k and that v is the velocity vector of
the line.
r(t) = ______i + _____j + _____k
r(t) = (-5 + 5t)i + (5t)j + (-2 + 2t)k
The work
6. Concider the line that passes through the point P(-2, -5, -5),
and is parallel to the line x = 1+2t, y = 2+4t, z = 3+3t
Find the point of intersection of this new line with each of
the coordinate planes:
xy-plane: (______, _____, _______)
xz-plane: (______, _____, _______)
yz-plane: (______, _____, _______)
xy-plane: ( 4/3, 5/3, 0)
xz-plane: ( 0.5, 0, -5/4)
yz-plane: ( 0, -1, -2)
The work
7. Find the vector and parametric equations for the line through the
point P(-3, 0, -2) and orthogonal to the plane -1x+2y+5z = 4.
Vector Form: r = < ______, ______, -2> + t< ______, ______, 5>
Parametric form (parameter t, and passing through P when t
= 0):
x = x(t) = __________
y = y(t) = __________
z = z(t) = __________
r = < -3,  0, -2> + t< -1, 2, 5>

x = x(t) = -3 - t
y = y(t) = 2t
z = z(t) = -2 + 5t
The work
8. Find the vector and parametric equations for the line through the
point P(-3, 5, 1) and the point Q(2, 2, -3).
Vector Form: r = < ____, _____, 1 > + t<  _____, _____, -4>
Parametric form (parameter t, and passing through P when t
= 0):
x = x(t) = _________
y = y(t) = _________
z = z(t) = _________
r = < -3, 5, 1 > + t<  5, -3, -4>

x = x(t) =-3 + 5t
y = y(t) =5 -3t
z = z(t) = 1 - 4t
The work
9. Find the vector equation for the line of intersection of the planes
x - 2y + 4z = -5 and x + 5z = -4
r = < _____, ______,0 > + t< -10, _____, ______ >
r = < -4, 0.5, 0 > + t< -10, -1, 2 >
The work
10. Consider the two lines
L1 : x = -2t, y = 1+2t, z = 3t and
L2 : x = -9 + 5s, y = 1+4s, z = 2 + 4s
Find the point of intersection of the two lines.
P = ( _____, ______, ______)
P = ( -4, 5, 6)
The work
11. Find an equation of the plane orthogonal to the line
(x, y, z) = (8,-5, -8) + t(-7, -10, -9)
which passes through the point (-5, -8, 8).
Give your answer in the form ax+by+cz = d (with a = 7).
a = _____
b = _____
c = _____
d = _____
a = 7
b = 10
c = 9
d = -43
The work
12. Find an equation of a plane through the point (-1, 0, -5) which
is orthogonal to the line
x = -5 - 3t, y = 2 - 1t, z = 5+5t
in which the coefficient of x is -3.
_______________________ = 0
-3(x + 1) - y + 5(z + 5) = 0
The work
13. Find an equation of a plane through the point (1, 4, 0) which
is parallel to the plane -2x - 1y + 4z=7 in which the coefficient
of x is -2.
______________________ = 0.
-2(x - 1) - (y - 4) + 4z
The work
14. Consider the plane which passes through the three points:
( -4, -4, 8) , (0, -6, 5), and (0;-5, 7).
Find the vector normal to this plane which has the form:
( 1, ______, ______ )
( 1, 8, -4 )
The work
15. Find an equation of a plane containing the three points (-2,
-2, -5), (0, 1, 0), (0, 2, 2) in which the coefficient of x is 1.
_____________________ = 0.
(x + 2) - 4(y + 2) + 2(z + 5) = 0
The work
16. Find the point at which the line < -3, 2, 1 > + t< 5, 3, -5> intersects
the plane 2x - 3y - 2z = 8.
( _______, ________, _________)
( 7,  8,  -9 )
The work
17. Find the point P where the line x = 1 + t, y = 2t,
z = -3t intersects the plane x + y - z = -5.
P = ( ______, ________, ________)
P = ( 0,  -2, 3)
The work
18. Find the angle of intersection of the plane 5x - 2y + 3z = 2 with
the plane 1x + 5y - 5z = -2.
Answer in radians:
_____________________
and in degrees:
_____________________
radians: 1.0991978578314

degrees: 62.9793981755246
The work
19. A vertical tower was built on a horizontal plane.  Several years
later the ground that the tower was built on had tilted and
measurements taken of 3 points on that ground gave
coordinates of (0, 0, 0), (3, 1, 0), and (0, 2, 3).  By what angle
does the tower deviate from the original vertical position?
___________ radians.
1.00685368543427 radians
The work
20. Consider the planes 2x + 2y + 1z = 1 and 2x + 1z = 0.
(A) Find the unique point P on the y-axis which is on both
planes. ( _______, _______, _______ )
(B) Find a unit vector u with positive first coordinate that is
parallel to both planes.
________i + ________j + _______k
(C) Use parts (A) and (B) to find a vector equation for the
line of intersection of the two planes,r(t) =
________i + ________j + _______k
A) ( 0,  1/2, 0 )
B) 0.4472135955i + 0j + -0.894427191k
C) (t/sqrt(5)) i + (1/2) j - (2t/sqrt(5)) k
The work
21. A) Find a vector equation of the line through the point (-3, 2,
1) that is normal to the plane 5x + 3y - 5z = 6
r(t) = <  ______, ______, _______ >.
B) Find the value of t where the above line intersects the
given plane ________.
C) The point on the plane where the above line intersects the
given plane is
( ________, _______, _______ ) .
D) Use (C) to find the distance from the point (-3, 2, 1) to the
plane 5x + 3y - 5z = 6.
A) r(t) = <  -3 + 5t,  2 + 3t, 1 - 5t >
B) 0.338983050847458
C) ( -1.305085,  3.016949153, -0.694915 )
D) 2.60377821961648
The work
22. For the point P = (0, 7, 3) and the line
x = -6 - 4t, y = -1 - 7t, z = 7 - 4t:
A) Find the distance between P and an arbitrary point on the
line, in terms of the parameter t
d = _________.
B) The value of t that minimizes the distance function above
is _________
C) The point on the line that is closest to P is
( ______, _______, ________ ).
D) The distance P is to the above line is _________
A) d= sqrt((-6 + -4t)^2 + (-8 - 7t)^2 + (4 - 4t)^2)

B) -0.790123456790123

C) ( -2.83950617,  4.53086419,  10.16049383 )

D) 8.08901098808946
The work