Practice Lines and planes in space
Lines and planes in space can be described using points, direction vectors or equations.
These descriptions help you study intersections, angles and distances.
Exercises with solutions
A line in space is described by a position vector and a direction vector: g: x = p + t · u. A plane is described by a position vector and two spanning vectors, or by a coordinate equation. For an intersection you set the representations equal and solve the system.
Exercise 1
A line has position vector (1; 2; 3) and direction vector (2; 0; −1). Where is the point for t = 2?
Show solution and working
(5; 2; 1)
p + 2 · u = (1; 2; 3) + (4; 0; −2) = (5; 2; 1).
Exercise 2
How do you recognise that two lines are parallel?
Show solution and working
from their direction vectors
Two lines are parallel when their direction vectors are multiples of each other.
Exercise 3
How do you check whether a point lies on a line?
Show solution and working
by setting it equal and determining t
Set the point equal to p + t · u. If every coordinate gives the same t, the point lies on the line.
Exercise 4
What do you need at minimum to define a plane?
Show solution and working
a position vector and two non-parallel spanning vectors
With only one spanning vector you describe a line; two different spanning vectors span a whole plane.
Common mistake
Confusing the position vector with the direction vector. The position vector is a fixed point of the line, while the direction vector only gives the direction. And not checking all three coordinates when testing a point.
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