Intensive Algebra 2
Vectors Equations of Lines & Planes
Name______________________
March 23
Equations of Lines & Planes:
1. Equations of a line: we have met 3 forms of the equation of a line in 2 dimensions
y  mx  b
y  y1  m( x  x1 )
A line has a direction & passes through a given point
ax  by  c
Now, we can define the vector equation of a line as:
r  a  tb or
x, y  x1 , y1  t a, b
where r  x, y , a  x1 , y1 , b  a, b
t b
a
r
(a) Write the vector equation of the line which passes through the point (3,4) and is in the
direction of the vector 1, 2 .
(b) Express x & y as functions of t, then eliminate t from the two equations. Is the equation
what you expected?
(c) Write the vector equation of the line through the points (2,1) & ( 3,-4). Then repeat the
process in (b) to find the usual form of the equation.
Intensive Algebra 2
Platt/Knaus
(d) Now let’s extrapolate and see how similar the process is when considering the following.
Write the vector equation of the line through the points (1,2,-1) and ( 3,4,0).
(e) What would happen if you tried to get the “regular” (called Cartesian) equation of a line
in 3–D?
(f) If we start with a line 3x + 4y =12.
i. What is the slope of this line?
ii. What is the direction vector of this line?
iii. What is a vector perpendicular to this line?
iv. (4,0) is clearly a point on the line. What is true of the vector x  4, y for any other
point (x,y) that is on the line?
(g) 1, 1 is a vector PERPENDICULAR to a line, whose equation we wish to find.
(3,2) is a point on the line. Use the idea dot product to find the equation of the line.
Intensive Algebra 2
Platt/Knaus
2. Equation of a plane:
The orientation of a plane can be uniquely determined by a vector which is perpendicular to it. The
specific plane can then be determined by a point on it. Picture this before continuing...
(a) (1,1,1) is a point on a plane, 3, 1, 2 is a vector perpendicular to the plane.
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).
(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
Homework Exercises:
Given the points A, B and vectors a, b, c, d answer questions 1-12:
A(0, -5)
B(4, 1)
a = -4i + j
b = 2i – 3j
1. Find a + b
4. Find the vector AB
c = i + 2j – 5k
2. Find -3d + 2c
5. Find the length of AB
3. Find lbl
6. Find the vector 2BA
7. Write a vector parallel to b but pointing in the opposite direction:
8. Write a vector parallel to a but with length 27:
9. Write a vector perpendicular to b:
10. Find the angle between the vectors a and b.
11. a. Find a vector equation of the line passing through B in the direction of b.
b. Draw this line in 2-D space on the graph given:
c. Write this equation in Cartesian form:
d = 4i + 7k
Intensive Algebra 2
Platt/Knaus
12. a. Find the vector equation for the line passing through A in the direction of a.
b. Find the intersection point of this line with the line you found in #11.
- - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 13. Convert the following function, y = 6x + 3, into a vector equation.
14. A vector equation of a line is r  1, 2  t 2,3 .
Re-write the equation of this line in standard Cartesian form where ax + by = c.
15. Find the line passing through (-1, 4, 0) and (3, -1, 2). Write your answer in vector form where r
= p + td. Then, write your equation in Cartesian form.
Intensive Algebra 2
Platt/Knaus
16. Find the Cartesian equation of the line that passes through the point (2, 1) and is perpendicular
to the vector 4i  3 j
*Challenge* 17. In this question, a unit vector represents a displacement of 1 meter. A miniature
car moves in a straight line, starting at the point (2, 0). After t seconds, its position (x, y) is given by
the vector equation : r  2,0  t 0.7,1
a. How far is the car from the point (0, 0) after 2 seconds?
b. Find the speed of the car.
c. Obtain the equation of the car’s path in the form ax + by = c.
d. If another car leaves at the same time as the miniature car and after t seconds its position is
given by the vector equation
, will the two cars collide? Why or why not.
Show your work for credit (a yes or no answer will get you nothing without the work to back it
up).
Intensive Algebra 2
Platt/Knaus