Lines and Planes

Question 1

what is a linear equation in x, y and z?

Answer 1

ax + by + cz = d

Question 2

how does an equation in x, y and z related to the points in R³?

Answer 2

every point (P) in R³ has 3 coordinates (x₀, y₀, z₀)

ax + by + cz = d

x = 1st coordinate

y = 2nd coordinate

z = 3rd coordinate

Question 3

how do you find a point on a plane given by an equation?

Answer 3

ex) 2x - y + z = 2
1. plug in any 2 number for the 2 variables in the equation
2. solve the equation for the 3rd variable

Question 4

normal vector

Answer 4

is a vector which is perpendicular to all directed line segments on the plane

Question 5

point normal form

Answer 5

to get an equation of the plane, we need:

1. normal vector
2. a point on the plane

n*(x-p) = 0

Question 6

equation of a plane standard form

Answer 6

ax + by + cz = d

standard form (a, b, c) is equal to the normal vector

Question 7

plane determined by 3 points

Answer 7

• normal vector is missing
• 3 points are given

normal vector (n) = (Q-P) x (R-P)

Question 8

distance of a point from a plane

Answer 8

Question 9

point-parallel form equation

Answer 9

to write an equation of a line

1. a point (P) on the line
2. a vector (V) which is parallel to the line

X(t) = P + tV

Question 10

how can we use the point-parallel form to find points on the line?

Answer 10

the variable t, which is called parameter, gives us different points on the line

set t to be any real number -> point on the line

Question 11

two-point form equation

Answer 11

1. two points (P) (Q) on a line
   - the parallel vector is unknown

X(t) = (1-t)P + tQ

Question 12

equations for lines in R²

Answer 12

Question 13

Find equations (in all five forms) for the line in R² which is through (1, -1) and is perpendicular to vector (2, 3)

Answer 13

Question 14

find equation in point-normal form for the plane through P(1,-2,-3) and is perpendicular to vector (2,1,-1)

Answer 14

Question 15

how to find the point of intersection of a line and plane

Answer 15

how to find the point of intersection:

1. find parametric equations for the line
2. plug in the parametric equations into the equation of the plane
3. solve the equation for (t) found in (step 2) and find the parameter (t)
4. plug in the value of the parameter (t) found in (step 3), into the parametric equation of the line. this way you find the coordinates of the points of intersection

Question 16

find the point of intersection of line X(t) = (1,3,4) + t(2,1,1) and plane

x + 3y - 2z = 8

Answer 16

Question 17

point of intersection of 2 lines

Answer 17

Question 18

how to find interseption of lines in parametric form (R² and R³)

Answer 18

1. make sure to use different parameters in the parametric equations of the line (use t in one end and s in the other)
2. set the corresponding coordinates of the 2 lines equal to each other
   ex) x(t) = x(s)

y(t) = y(s)

3. solve the equation for (s) or (t)
4. plug the parameter value found in step 3, into the corresponding equation to find the point

Question 19

find the point of intersection of lines x = 2t + 1, y = 3t - 2 and

X(s) = (3,5) + t(-2,2)

Answer 19