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Lines and planes in space
This test covers the main relationships between lines and planes in space. Tasks apply axioms of solid geometry, identify parallelism and perpendicularity and calculate distances and angles in three-dimensional figures.
Questions: 10Estimated time: 22 minutes
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Question 1out of 10
What relative positions can a line and a plane have?
Three noncollinear points determine one plane. A line may lie in it, intersect once or have no common point and be parallel. A line outside a plane that is parallel to a line in that plane is parallel to the plane.
Question 2out of 10
How many planes pass through three non-collinear points?
Three noncollinear points determine one plane. A line may lie in it, intersect once or have no common point and be parallel. A line outside a plane that is parallel to a line in that plane is parallel to the plane.
Question 3out of 10
The normal vectors of two planes are n₁ = (1; 0; 0) and n₂ = (5; 12; 0). Find the cosine of the acute angle between the planes.
The angle between two planes equals the acute angle between their normal vectors. Use cos α = |n₁ · n₂| / (|n₁||n₂|): calculate the dot product and the lengths of both vectors.
Question 4out of 10
What is a dihedral angle?
A dihedral angle is formed by two half-planes sharing an edge; measure it in a section perpendicular to the edge. The angle between a line and a plane is measured between the line and its orthogonal projection onto the plane.
Question 5out of 10
Which statement is a criterion for a line parallel to a plane?
Three noncollinear points determine one plane. A line may lie in it, intersect once or have no common point and be parallel. A line outside a plane that is parallel to a line in that plane is parallel to the plane.
Question 6out of 10
An oblique segment of length 5 has an orthogonal projection of length 4 on a plane. Find the sine of the angle α between the segment and the plane.
The oblique segment, its orthogonal projection and the perpendicular to the plane form a right triangle. First find the perpendicular length using the Pythagorean theorem. The sine of the angle between the segment and the plane is the perpendicular length divided by the oblique length.
Question 7out of 10
An oblique segment from point A to a plane is 25 cm long, and its orthogonal projection onto the plane is 24 cm. Find the distance from point A to the plane.
The distance from a point to a plane is the length of the perpendicular. The perpendicular, the oblique segment and its projection form a right triangle, with the oblique segment as the hypotenuse. Apply the Pythagorean theorem: if the oblique length is l and the projection is p, then h = √(l² − p²).
Question 8out of 10
Given plane α: 5x + 3y + 5z = −19 and line l: x = −3 + t; y = −1 + 3t; z = −2 − t, t ∈ ℝ, find the value of t at their intersection.
Substitute the line coordinates into the plane equation and solve the linear equation for t. The question asks for the parameter, not the intersection coordinates.
Question 9out of 10
How is the angle between an oblique line and a plane found?
A dihedral angle is formed by two half-planes sharing an edge; measure it in a section perpendicular to the edge. The angle between a line and a plane is measured between the line and its orthogonal projection onto the plane.
Question 10out of 10
Find the distance between the parallel planes 3x + 4y − 3 = 0 and 3x + 4y + 22 = 0.
The distance between the parallel planes Ax + By + Cz + D₁ = 0 and Ax + By + Cz + D₂ = 0 is |D₂ − D₁| / √(A² + B² + C²).
Time left: 00:00:00
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Sample questions
- How many planes pass through three non-collinear points?
- What relative positions can a line and a plane have?
- Which statement is a criterion for a line parallel to a plane?
- When is a line perpendicular to a plane?
- Find the distance from a point to a plane when the lengths of an oblique segment and its orthogonal projection are known.
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