Math Exercises: Relative Position, Distance and Deviation Between Points, Lines and Planes 
Find the distance between two points :
Find the distance from a point to a straight line :
Find the distance from a point to a plane :
Determine the relative position of the given straight lines, calculate the angle between them and find the intersection of the straight lines (if any exists) :
Determine the relative position of the straight line and the plane, calculate the angle between them and find their intersection (if any exists) :
Determine the relative position of the given planes, calculate the angle between them and find the intersection of the planes (if any exists) :
Determine the relative position of three planes :
Find the distance between two straight lines p: 3x – 4y – 20 = 0 and q: 6x – 8y + 25 = 0.
Find the distance between a straight line p: {x = 2t – 1; y = 1 – t; z = 2 + 3t; t∈R} and a plane
Find the distance between two planes α: 2x + y + 3z + 1 = 0 and β: 6x + 3y + 9z + 5 = 0.
Find the general equation of a straight line that passes through the point M [15;–3] and through the intersection of the straight lines p: 3x – 5y + 12 = 0 and q: 5x + 2y – 42 = 0.
Find the general equation of a straight line that passes through the point A [3;–2], if the size of an angle between the unknown straight line and the straight line p: x – y + 1 = 0 is α = 30°.
Find the general equation of a straight line that passes through the point A [2;3], if the distance from a point B [0;–1] to the unknown straight line is d = 4.
Find the image of a point A [1;0;2] under the plane symmetry given by the plane β: x – 2y + 3z – 21 = 0.
Two sides of a parallelogram lie on straight lines 8x + 3y + 1 = 0, 2x + y – 1 = 0 and the diagonal of a parallelogram lies on a straight line 3x + 2y + 3 = 0. Find the coordinates of vertices of a parallelogram.
Find the size of internal angles of the triangle ABC, if A [4;0;6], B [6;–3;12], C [10;2;3].
Two vertices of a triangle ABC have coordinates A [–10;2], B [6;4] and the orthocenter of a triangle is O [5;2]. Find the coordinates of a vertex C.
Sides of a triangle lie on straight lines a: 3x + 4y – 1 = 0, b: x – 7y – 17 = 0, c: 7x + y + 31 = 0. Find the coordinates of vertices A, B, C of a triangle.
Vertices of a regular tetrahedron have the coordinates A [6;0;0], B [0;5;0], C [5;6;0], D [2;3;8]. Find the angle between straight lines AB, CD and the angle between a straight line CD and a plane ABD.
Consider a regular square pyramid ABCDE whose base lies on the xyplane, A [0;0;0], B [5;0;0], D [0;5;0] and the vertical height of a pyramid h = 7.
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