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MTH 212 Calculus III
Notes 11.5 Lines and Planes in Space
Consider a line in space through the point P( x1 , y1 , z1 ), in the direction of v = <a, b, c> (the
direction vector). If the point Q (x, y, z) is on this line, then the vector PQ must be parallel with the
direction vector v.
Thus we can write:
PQ = t v
for some scalar t.
This means:
PQ =  x  x1 , y  y1 , z  z1 
  at , bt , ct 
Since these two vectors are equal, their corresponding components are equal, and we obtain the
parametric equations of a line in space:
x  x1  at,
y  y1  bt,
z  z1  ct
for t a real number.
If you solve each of these parametric equations for t, and the direction numbers a, b, and c are nonzero,
we can write the symmetric equations of a line in space:
Ex. Find a set of parametric equations and the corresponding set of symmetric equations of the line
that passes through the points (–5, 2, 3) & (3, 1, 0).
Planes in Space
The equation of a plane can be obtained from a point on the plane and a vector normal (perpendicular)
to it.
Consider a plane containing the point P( x1 , y1 , z1 ), with the normal vector n = <a, b, c>.
Then the plane consists of all points Q(x, y, z) such that the vector PQ is orthogonal to n.
This means that for any point Q(x, y, z) on the plane,
n · PQ = 0
Ex. Find the equation of the plane containing the points (2, 3, -2), (3, 4, 2) & (1, -1, 0).