Intensive Algebra 2
Eqns of Planes & Other Practice
Name______________________
Mar. 30/31
Vectors & Equations of a plane:
The orientation of a plane can be uniquely determined by a vector which is perpendicular to it,
called a normal vector. The specific plane can then be determined by a point on it. Picture this
before continuing...
(a) If (1,1,1) is a point on a plane and 3, 1, 2 is the plane’s normal vector, then
x  1, y  1, z  1 is a vector lying on the plane for any (x,y,z) on the plane. Use the dot
product to find the equation of this plane.
(b) 3x + 2y – z = 0 is the equation of the plane. What can you say about this plane?
(c) Find the equation of the plane which is the PERPENDICULAR BISECTOR of the line
segment joining the points ( 2,3,-4) & (4,-5,8). (hint: draw a picture)
(d) x, y, z  1, 1,1  t 2, 3,1 is a vector equation of a line in space.
2x - 3y + z = 6 is the equation of a plane. What is the relationship between the line and the
plane?
(e) x + y + z = 1 is another plane. What is the relationship between the line in (d) and this
plane?
Intensive Algebra 2
Platt/Knaus
Practice bringing it all together: (you may use your calculator unless otherwise noted)
1. (a) Give the vector equation of a line joining the points A(1,-1,3) and B( -2,3,5)
(b) Find the equation of the plane which is perpendicular to this line and passes through the
point A.
x  2 y  4 z  12
2. Given the three planes:
2x  3y  z  3
3x  4 y  2 z  6
(a) Find the intersection of the three planes using matrices and row operations (i.e. with an
augmented matrix)
(b) Rewrite the system as a matrix equation:
(c) Solve the system using an inverse matrix
(d) Are any of the planes parallel or perpendicular to each other? Explain how you arrived
at your answer. (use your knowledge of planes and their normal vector!)
x  2 y  4z  4
Intensive Algebra 2
Platt/Knaus
3. Given the three planes: 2 x  3 y  z  3
3x  y  5 z  7
(a) Write the system as a matrix equation
(b) Find the determinant of the 3x3 matrix.
(c) What does the value of the determinant tell you about the solution of the system?
(d) Does the system have any solutions?
(e) Find the complete solution of the system writing your answer in the form of a vector
equation.
(f) What is the relationship between the rows of the matrix?
Intensive Algebra 2
Platt/Knaus
4. Given real numbers w, x, y and z, and the following matrices:
2 3 
A

 w 1
 1 x 1 
B

 0 3 4
 3 2 
C  1 0 


 y 4 
 1 0 4
D   1 z 2 


 0 1 3 
Carry out each indicated operation, or explain why it cannot be performed.
a) AB
b) C
2
1
c) A
d) B
1
e) BC  A
5. Assuming A and D are invertible, solve the following equation for X , where X is a
4 x 1variable matrix and state the dimensions of all other matrices, where B is also a
4 x 1variable matrix.
DAX  C  B
Intensive Algebra 2
Platt/Knaus
6. An encyclopedia saleswoman works for a company that offers three different grades of bindings
for its encyclopedias: standard, deluxe, leather. For each set she sells, she earns a commission
that is based on the set’s binding grade. One week she sells one standard, one deluxe, and two
leather sets and makes $675 in commission. The next week she sells two standard, one deluxe,
and one leather set for a $600 commission. The third week she sells one standard, two deluxe,
and one leather set earning $625 in commission.
a. Let x, y, z represent the commission she earns on standard, deluxe, and leather sets,
respectively. Express the system of equations as a matrix equation.
b. Solve the system and express, in words, the meaning of each of the solution.
7. For what value of k does the following system have infinitely many solutions?
kx  y  z  0
x  2 y  kz  0
 x  2 y  3z  0
Intensive Algebra 2
Platt/Knaus
8. The vector <1, 1, 1> is perpendicular to a plane that contains the point (3, 4, 5). Write the
equation for this plane.
9. Find a vector perpendicular to the planes:
a. 3x – 5y + 4z = 0
b. x – z =2
c. z = 1
10. Consider the points A (2, 2, 2) and B (4, 6, 8)
a. Find a Cartesian equation of the plane that is perpendicular to vector AB at the
midpoint of A and B.
b. Show that the point P (2, 0, 8) is on the plane.
11. State which of the following planes are parallel or perpendicular, give justification.
P1 : 3x  2 y  z  6
P2 : 6 x  4 y  2 z  8
P3 : 4 x  2 y  8 z  7
12. Explain how you know that the line <x, y, z> = <3, 1, 4> + t<4, -5, 2> is parallel to the plane
2x + 2y + z = 7.