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: xyplane: (______, _____, _______) xzplane: (______, _____, _______) yzplane: (______, _____, _______) 
xyplane: ( 4/3, 5/3, 0) xzplane: ( 0.5, 0, 5/4) yzplane: ( 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 yaxis 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 
