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*: 3*x* – 4*y* – 20* = *0 and *q*: 6*x* – 8*y* + 25* = *0.

Find the distance between a straight line *p*: {*x* = 2*t* – 1; *y* = 1 – *t*; *z* = 2 + 3*t*; *t*∈**R**} and a plane*ρ*: *x* + 5*y* + *z* – 3 = 0.

Find the distance between two planes α: 2*x* + *y* + 3*z* + 1 = 0 and *β*: 6*x* + 3*y* + 9*z* + 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*: 3*x* – 5*y* + 12 = 0 and *q*: 5*x* + 2*y* – 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* – 2*y* + 3*z* – 21 = 0.

Two sides of a parallelogram lie on straight lines 8*x* + 3*y* + 1 = 0, 2*x* + *y* – 1 = 0 and the diagonal of a parallelogram lies on a straight line 3*x* + 2*y* + 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*: 3*x* + 4*y* – 1 = 0, *b*: *x* – 7*y* – 17 = 0, *c*: 7*x* + *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 *xy-*plane, *A* [0;0;0], *B* [5;0;0], *D* [0;5;0] and the vertical height of a pyramid *h = *7.

**a)** Find the distance from a point *A* to a point *C*.

**b)** Find the distance from a point *A* to a point *E*.

**c)** Find the distance from a point *E* to a midpoint of the edge *AB*.

**d)** Find the distance between midpoints of the edges *AE* and *CE*.

**e)** Find the size of an angle between adjacent faces of the pyramid.

**f)** Find the size of an angle between opposite faces of the pyramid.

**g)** Find the size of an angle between straight lines *BC* and *DE*.

**h)** Find the size of an angle between straight lines *BE* and *DE*.

**i)** Find the size of an angle between a straight line *AE* and a plane *ABC*.

**j)** Find the size of an angle between a straight line *AE* and a plane *BCE*.

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