Analytic geometry. Textbook

To nd canonical equations of this line we note that its direction vector is orthogonal to the normal vectors ~n1 = fA1; B1; C1g and ~n2 = fA2; B2; C2g of these planes. Therefore we may take the vector ~n1 ~n2 as the direction vector. Taking any solution of this system of linear equations we get a point of this straight line.

Definition. Two straight lines in the space are called parallel if they lie in the same plane and do not intersect or coincide.

Let us consider a straight line `1 with a direction vector ~e1 = fk1; l1; m1g which passes through a point M1(x1; y1; z1), and a straight line `2 de ned similarly by a vector ~e2 = fk2; l2; m2g and a point M2(x2; y2; z2). These straight lines have the standard equations

The lines `1 and `2 are parallel if and only if their direction vectors are collinear. Therefore these lines are parallel if and only if

k1 = l1 = m1 : k2 l2 m2

Equations (9) de ne the same straight line if and only if these lines are parallel and the point M2 is on the line `1 (or, equiavalently the point M1 is on the line `2). Therefore these equations de ne the same straight line if and only if

Now we nd when the straight lines de ned by the equations (9) lie on

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one plane. This property holds if and only if the vectors M1M2, ~e1 and ~e2 are coplanar (see Fig. 16).

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Writing out the condition of coplanarity M1M2 ~e1 ~e2 = 0 we get that

x

Fig. 16. The straight lines lie on one plane

We assume that this condition is valid and nd equation of the plane con-

taining these straight lines. The normal vector ~n of the desired plane is

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orthogonal to each of the vectors M1M2, ~e1, ~e2.

desired plane:

x x1 y y1 z z1

If the straight lines `1 and `2 are parallel, then their direction vectors ~e1

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and ~e2 are collinear, but the vector M1M2 is not collinear with these vectors

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since otherwise `1 = `2. We may take ~n = M1M2 ~e1 and using similar arguments as in the previous case we get the equation of the plane in the

Definition. The straight lines are called orthogonal if their direction vectors are orthogonal.

So the straight lines de ned by the equations (9) are orthogonal if and only if k1k2 + l1l2 + m1m2 = 0.

Suppose that the straight line and plane are determined respectively by the following equations

We are going to nd when these line and plane are orthogonal, parallel or the straight line lies on the plane.

The straight line is orthogonal to the plane if and only if the direction vector ~e = fk; l; mg and the normal vector of the plane ~n = fA; B; Cg are collinear. Therefore we get the following condition

Ak = Bl = mC :

z

`

~n ~e

O

y

x

Fig. 17. The straight line is orthogonal to the plane

The given straight line is parallel to the plane if and only if the direction vector ~e = fk; l; mg and the normal vector of the plane ~n = fA; B; Cg are orthogonal.

z

`

~e

~n

O

y

x

Fig. 18. The straight line is parallel to the plane

Therefore we get the following condition Ak + Bl + Cm = 0.

The straight line lies on the plane if and only if the point M0(x0; y0; z0) is on the plane and the line is parallel to the plane, so we get the following conditions

Ax0 + By0 + Cz0 + D = 0; Ak + Bl + Cm = 0:

Theorem 8. The distance d of the point M1 from the line with the direction vector ~e passing through the point M0 may be found by the formula

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d = jM0M1 ~e j: j~e j

Proof. We assume that M0 is the initial point of ~e. Let us considerrst the case when the point M1 is not on the line.

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We consider the parallelogram de ned by the vectors MM0 and ~e and

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take MM0 as the base of this parallelogram. Then distance of M1 from the given line equals the height of this parallelogram. The area A of the

parallelogram is obtained by the formula A = j~e j d, the length of the base is j~e j, and from the relation A = j~e j d we get that

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M M and ~e are collinear, therefore M M = 0 and the formula gives the valid value d = 0.

Corollary. The distance between two parallel straight lines passing through the points M1 and M2 respectively and having the same direction vector ~e can be found from the equality

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d = jM1M2 ~e j: j~e j

Proof. The distance d between these straight lines equals the distance of the point M2 from the straight line passing through the point M1. Using the formula obtained in the theorem we get the desired formula.

Definition. Two straight lines are called skew lines if they do not intersect and are not parallel.

Remark. Two straight lines are skew if and only if they do not lie in one plane.

Theorem 9. Let `1 and `2 be skew straight lines de ned by the points

M1, M2 and the direction vectors ~e1, ~e2 respectively. Then the distance d between these lines may be de ned be the equality

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d = jM1M2 ~e1 ~e2j: j~e1 ~e2j

Proof. We assume that the vectors are reduced to the initial point M1.

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Three vectors ~e1, ~e2 and M1M2 de ne a parallelepiped. We consider the parallelogram de ned be the vectors ~e1 and ~e2 as the base of this parallelepiped. Then the desired distance d equals the height of the parallelepiped.

`2

M2

Let V be the volume of this parallelepiped and A be the area of its base.

Remark. The formula obtained in the previous theorem remains valid if the given lines are not parallel (regardless of presence of their intersection

point). It remains to consider the case when these straight lines lie in one plane. In this case d = 0 and the formula gives the correct value since

3.4Some tasks on straight lines and planes in the space

1)Find the equation of the straight line passing through the points

M1(x1; y1; z1) and M2(x2; y2; z2), M1 6= M2.

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Fig. 21. The straight line de ned by two points

From the relations

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M1M = fx x1; y y1; z z1g;

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M1M2 = fx2 x1; y2 y1; z2 z1g;

we get equation of the desired straight line

2) Find the equation of the plane passing through the point M0(x0; y0; z0) and perpendicular to the planes

A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0:

It is assumed that the given planes are not parallel.

The normal vector ~n of the desired plane is orthogonal to the normal vectors ~n1 = fA1; B1; C1g and ~n2 = fA2; B2; C2g. We may take ~n = ~n1 ~n2

points do not lie on some straight line1).

x

Fig. 22. Plane de ned by three non-collinear points

The point M(x; y; z) lies on the plane de ned by three given points if

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and only if the vectors M0M, M0M1 and M0M2 are coplanar. The equation

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of this plane takes the form M0M M0M1 M0M2 = 0.

1) The points M0, M1, . . . , Mk are called collinear if they all lie on one straight line. It is obvious that

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the points M0, M1, . . . , Mk are collinear if and only if the vectors M1M0, M2M0, . . . , MkM0 are collinear.

Taking into account that

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M0M = fx x0; y y0; z z0g;

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M0M1 = fx1 x0; y1 y0; z1 z0g;

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M0M2 = fx2 x0; y2 y0; z1 z0g;

we rewrite equation of the plane in the coordinate form