Lines and planes in space · Grade 12 · 5 units
In Lines and planes in space (Grade 12 · 5 units) you practise 6 skills: The line in space · How two lines lie · The equation of a plane · A line and a plane · Distances in space · Angles between planes. A line in space is fixed by one point on it and a direction vector. Every point of the line is the given point plus a multiple of the direction, and the number you multiply by is called the parameter. To check whether a point is on a line, look for one parameter that gives all three of its coordinates. Two lines in space can be parallel, the same line, intersecting or skew. If the directions are parallel, the lines are parallel or the same line. Otherwise set the coordinates equal: if there is a solution the lines intersect, and if there is none they are skew.
The equation of a plane has degree one in three variables. The coefficients of the variables are the components of a normal vector, a vector perpendicular to every line of the plane. To find the plane through three points, look for a normal perpendicular to two vectors between the points, then put in a point to find the constant. Two planes that are not parallel meet in a line, and its direction is perpendicular to both normals. A line meets a plane at one point when its direction is not perpendicular to the normal. If it is perpendicular, the line is parallel to the plane or lies in it, and one point of the line decides which. The distance of a point from a plane is the absolute value of what you get when you put the point in the equation, divided by the size of the normal. The same idea gives the distance between parallel planes and between skew lines: take the plane through the first line that is parallel to the second, and the distance between the lines is the distance of a point of the second line from that plane. The angle between two planes is the angle between their normals that is not obtuse, so its cosine is the absolute value of the scalar product of the normals, divided by the product of their sizes. The angle between a line and a plane and the angle between the line and the normal add up to ninety degrees, so the sine of the first is found the same way.
Tip: Before you work out the angle between a line and a plane, remember that the formula with the normal gives the sine, not the cosine.
What you learn
The line in space
For example: On the line x⃗ = (0, 5, 1) + t(3, 4, −1) , take the point where t = −1 . What is the x coordinate of that point?
Answer and solution
The answer: −3
Substitute t = −1 into the x coordinate: x = 0 + (−1) × 3 , that is, x = −3 .
How two lines lie
For example: The lines x⃗ = (0, 3, 5) + t(0, 2, 1) and x⃗ = (3, 2, 2) + k(0, 4, 2) are given. How do the two lines lie?
- Intersecting
- Skew
- The same line
- Parallel
Answer and solution
The answer: Parallel
The directions are (0, 2, 1) and (0, 4, 2) . The directions are parallel, and the given point of the second line is not on the first, so the lines are parallel.
The equation of a plane
For example: The plane −5x + 6y − 5z + d = 0 passes through the point P(0, 4, 2) . What is d ?
Answer and solution
The answer: −14
Substitute the coordinates of the point in the equation: 14 + d = 0 , so d = −14 .
A line and a plane
For example: The line x⃗ = (0, 5, 4) + t(3, 2, 0) and the plane 2x + 3y − 3z + 9 = 0 are given. For which value of t does the line meet the plane?
Answer and solution
The answer: −1
Putting in the general point gives 12 + 12t = 0 , so t = −1 .
Distances in space
For example: What is the distance of the point P(−3, −3, −4) from the plane 2x − y − 2z + 1 = 0 ?
Answer and solution
The answer: 2
Substituting the point gives 6 , and the size of the normal is √9 = 3 , so the distance is 6 / 3 = 2 .
Angles between planes
For example: What is the angle between the planes y − z + 2 = 0 and 2x − y − z − 5 = 0 ?
Answer and solution
The answer: 90 °
The normals are (0, 1, −1) and (2, −1, −1) , so cos α = 0 / √12 and α = 90° .
Not on the site yet
Parts of this topic that are not built yet: The parametric form of a plane · The distance between skew lines by the geometric method.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).