Lines and Planes Flashcards
what is a linear equation in x, y and z?
ax + by + cz = d
how does an equation in x, y and z related to the points in R3?
every point (P) in R3 has 3 coordinates (x0, y0, z0)
ax + by + cz = d
x = 1st coordinate
y = 2nd coordinate
z = 3rd coordinate
how do you find a point on a plane given by an equation?
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
normal vector
is a vector which is perpendicular to all directed line segments on the plane
point normal form
to get an equation of the plane, we need:
- normal vector
- a point on the plane
n*(x-p) = 0
equation of a plane standard form
ax + by + cz = d
standard form (a, b, c) is equal to the normal vector
plane determined by 3 points
- normal vector is missing
- 3 points are given
normal vector (n) = (Q-P) x (R-P)
distance of a point from a plane
point-parallel form equation
to write an equation of a line
- a point (P) on the line
- a vector (V) which is parallel to the line
X(t) = P + tV
how can we use the point-parallel form to find points on the line?
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
two-point form equation
- two points (P) (Q) on a line
- the parallel vector is unknown
X(t) = (1-t)P + tQ
equations for lines in R2
Find equations (in all five forms) for the line in R2 which is through (1, -1) and is perpendicular to vector (2, 3)
find equation in point-normal form for the plane through P(1,-2,-3) and is perpendicular to vector (2,1,-1)
how to find the point of intersection of a line and plane
how to find the point of intersection:
- find parametric equations for the line
- plug in the parametric equations into the equation of the plane
- solve the equation for (t) found in (step 2) and find the parameter (t)
- 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
find the point of intersection of line X(t) = (1,3,4) + t(2,1,1) and plane
x + 3y - 2z = 8
point of intersection of 2 lines
how to find interseption of lines in parametric form (R2 and R3)
- make sure to use different parameters in the parametric equations of the line (use t in one end and s in the other)
- set the corresponding coordinates of the 2 lines equal to each other
ex) x(t) = x(s)
y(t) = y(s)
- solve the equation for (s) or (t)
- plug the parameter value found in step 3, into the corresponding equation to find the point
find the point of intersection of lines x = 2t + 1, y = 3t - 2 and
X(s) = (3,5) + t(-2,2)
