Problem 1
Find the cosine of the angle between the lines $\frac(x+3)(5) =\frac(y-2)(-3) =\frac(z-1)(4) $ and $\left\(\begin(array )(c) (x=2\cdot t-3) \\ (y=-t+1) \\ (z=3\cdot t+5) \end(array)\right. $.
Let two lines be given in space: $\frac(x-x_(1) )(m_(1) ) =\frac(y-y_(1) )(n_(1) ) =\frac(z-z_(1 ) )(p_(1) ) $ and $\frac(x-x_(2) )(m_(2) ) =\frac(y-y_(2) )(n_(2) ) =\frac(z- z_(2) )(p_(2) ) $. Let's choose an arbitrary point in space and draw through it two auxiliary lines parallel to the data. The angle between these lines is any of the two adjacent angles formed by the auxiliary lines. The cosine of one of the angles between straight lines can be found using the well-known formula $\cos \phi =\frac(m_(1) \cdot m_(2) +n_(1) \cdot n_(2) +p_(1) \cdot p_( 2) )(\sqrt(m_(1)^(2) +n_(1)^(2) +p_(1)^(2) ) \cdot \sqrt(m_(2)^(2) +n_( 2)^(2) +p_(2)^(2) ) ) $. If the value $\cos \phi >0$, then we get sharp corner between lines if $\cos \phi
Canonical equations of the first line: $\frac(x+3)(5) =\frac(y-2)(-3) =\frac(z-1)(4) $.
The canonical equations of the second line can be obtained from the parametric ones:
\ \ \
Thus, the canonical equations of this line are: $\frac(x+3)(2) =\frac(y-1)(-1) =\frac(z-5)(3) $.
We calculate:
\[\cos \phi =\frac(5\cdot 2+\left(-3\right)\cdot \left(-1\right)+4\cdot 3)(\sqrt(5^(2) +\ left(-3\right)^(2) +4^(2) ) \cdot \sqrt(2^(2) +\left(-1\right)^(2) +3^(2) ) = \frac(25)(\sqrt(50) \cdot \sqrt(14) ) \approx 0.9449.\]
Problem 2
The first line passes through the given points $A\left(2,-4,-1\right)$ and $B\left(-3,5,6\right)$, the second line passes through the given points $C\left (1,-2,8\right)$ and $D\left(6,7,-2\right)$. Find the distance between these lines.
Let a certain line be perpendicular to the lines $AB$ and $CD$ and intersect them at points $M$ and $N$, respectively. Under these conditions, the length of the segment $MN$ is equal to the distance between the lines $AB$ and $CD$.
We construct the vector $\overline(AB)$:
\[\overline(AB)=\left(-3-2\right)\cdot \bar(i)+\left(5-\left(-4\right)\right)\cdot \bar(j)+ \left(6-\left(-1\right)\right)\cdot \bar(k)=-5\cdot \bar(i)+9\cdot \bar(j)+7\cdot \bar(k ).\]
Let the segment depicting the distance between the lines pass through the point $M\left(x_(M) ,y_(M) ,z_(M) \right)$ on the line $AB$.
We construct the vector $\overline(AM)$:
\[\overline(AM)=\left(x_(M) -2\right)\cdot \bar(i)+\left(y_(M) -\left(-4\right)\right)\cdot \ bar(j)+\left(z_(M) -\left(-1\right)\right)\cdot \bar(k)=\] \[=\left(x_(M) -2\right)\ cdot \bar(i)+\left(y_(M) +4\right)\cdot \bar(j)+\left(z_(M) +1\right)\cdot \bar(k).\]
The vectors $\overline(AB)$ and $\overline(AM)$ are the same, therefore they are collinear.
It is known that if the vectors $\overline(a)=x_(1) \cdot \overline(i)+y_(1) \cdot \overline(j)+z_(1) \cdot \overline(k)$ and $ \overline(b)=x_(2) \cdot \overline(i)+y_(2) \cdot \overline(j)+z_(2) \cdot \overline(k)$ are collinear, then their coordinates are proportional, then there is $\frac(x_((\it 2)) )((\it x)_((\it 1)) ) =\frac(y_((\it 2)) )((\it y)_( (\it 1)) ) =\frac(z_((\it 2)) )((\it z)_((\it 1)) ) $.
$\frac(x_(M) -2)(-5) =\frac(y_(M) +4)(9) =\frac(z_(M) +1)(7) =m$, where $m $ is the result of division.
From here we get: $x_(M) -2=-5\cdot m$; $y_(M) +4=9\cdot m$; $z_(M) +1=7\cdot m$.
We finally obtain expressions for the coordinates of point $M$:
We construct the vector $\overline(CD)$:
\[\overline(CD)=\left(6-1\right)\cdot \bar(i)+\left(7-\left(-2\right)\right)\cdot \bar(j)+\ left(-2-8\right)\cdot \bar(k)=5\cdot \bar(i)+9\cdot \bar(j)-10\cdot \bar(k).\]
Let the segment representing the distance between the lines pass through the point $N\left(x_(N) ,y_(N) ,z_(N) \right)$ on the line $CD$.
We construct the vector $\overline(CN)$:
\[\overline(CN)=\left(x_(N) -1\right)\cdot \bar(i)+\left(y_(N) -\left(-2\right)\right)\cdot \ bar(j)+\left(z_(N) -8\right)\cdot \bar(k)=\] \[=\left(x_(N) -1\right)\cdot \bar(i)+ \left(y_(N) +2\right)\cdot \bar(j)+\left(z_(N) -8\right)\cdot \bar(k).\]
The vectors $\overline(CD)$ and $\overline(CN)$ coincide, therefore, they are collinear. We apply the condition of collinearity of vectors:
$\frac(x_(N) -1)(5) =\frac(y_(N) +2)(9) =\frac(z_(N) -8)(-10) =n$, where $n $ is the result of division.
From here we get: $x_(N) -1=5\cdot n$; $y_(N) +2=9\cdot n$; $z_(N) -8=-10\cdot n$.
We finally obtain expressions for the coordinates of point $N$:
We construct the vector $\overline(MN)$:
\[\overline(MN)=\left(x_(N) -x_(M) \right)\cdot \bar(i)+\left(y_(N) -y_(M) \right)\cdot \bar (j)+\left(z_(N) -z_(M) \right)\cdot \bar(k).\]
We substitute expressions for the coordinates of points $M$ and $N$:
\[\overline(MN)=\left(1+5\cdot n-\left(2-5\cdot m\right)\right)\cdot \bar(i)+\] \[+\left(- 2+9\cdot n-\left(-4+9\cdot m\right)\right)\cdot \bar(j)+\left(8-10\cdot n-\left(-1+7\cdot m\right)\right)\cdot \bar(k).\]
Having completed the steps, we get:
\[\overline(MN)=\left(-1+5\cdot n+5\cdot m\right)\cdot \bar(i)+\left(2+9\cdot n-9\cdot m\right )\cdot \bar(j)+\left(9-10\cdot n-7\cdot m\right)\cdot \bar(k).\]
Since the lines $AB$ and $MN$ are perpendicular, the scalar product of the corresponding vectors is equal to zero, that is, $\overline(AB)\cdot \overline(MN)=0$:
\[-5\cdot \left(-1+5\cdot n+5\cdot m\right)+9\cdot \left(2+9\cdot n-9\cdot m\right)+7\cdot \ left(9-10\cdot n-7\cdot m\right)=0;\] \
Having completed the steps, we obtain the first equation for determining $m$ and $n$: $155\cdot m+14\cdot n=86$.
Since the lines $CD$ and $MN$ are perpendicular, the scalar product of the corresponding vectors is equal to zero, that is, $\overline(CD)\cdot \overline(MN)=0$:
\ \[-5+25\cdot n+25\cdot m+18+81\cdot n-81\cdot m-90+100\cdot n+70\cdot m=0.\]
Having completed the steps, we obtain the second equation for determining $m$ and $n$: $14\cdot m+206\cdot n=77$.
We find $m$ and $n$ by solving the system of equations $\left\(\begin(array)(c) (155\cdot m+14\cdot n=86) \\ (14\cdot m+206\cdot n =77)\end(array)\right.$.
We apply the Cramer method:
\[\Delta =\left|\begin(array)(cc) (155) & (14) \\ (14) & (206) \end(array)\right|=31734; \] \[\Delta _(m) =\left|\begin(array)(cc) (86) & (14) \\ (77) & (206) \end(array)\right|=16638; \] \[\Delta _(n) =\left|\begin(array)(cc) (155) & (86) \\ (14) & (77) \end(array)\right|=10731;\ ]\
Find the coordinates of points $M$ and $N$:
\ \
Finally:
Finally, we write the vector $\overline(MN)$:
$\overline(MN)=\left(2.691-\left(-0.6215\right)\right)\cdot \bar(i)+\left(1.0438-0.7187\right)\cdot \bar (j)+\left(4.618-2.6701\right)\cdot \bar(k)$ or $\overline(MN)=3.3125\cdot \bar(i)+0.3251\cdot \bar( j)+1.9479\cdot \bar(k)$.
The distance between lines $AB$ and $CD$ is the length of the vector $\overline(MN)$:$d=\sqrt(3.3125^(2) +0.3251^(2) +1.9479^( 2) ) \approx 3.8565$ lin. units
Let two straight lines l and m on a plane in Cartesian system coordinates are given general equations: l: A 1 x + B 1 y + C 1 = 0, m: A 2 x + B 2 y + C 2 = 0
Normal vectors to these lines: = (A 1 , B 1) – to line l,
= (A 2 , B 2) – to line m.
Let j be the angle between lines l and m.
Since angles with mutually perpendicular sides are either equal or add up to p, then 
, that is, cos j = .
So, we have proven the following theorem.
Theorem. Let j be the angle between two lines on the plane, and let these lines be specified in the Cartesian coordinate system by the general equations A 1 x + B 1 y + C 1 = 0 and A 2 x + B 2 y + C 2 = 0. Then cos j = 
.
Exercises.
1) Derive a formula for calculating the angle between straight lines if:
(1) both lines are specified parametrically; (2) both lines are given canonical equations; (3) one line is specified parametrically, the other line is specified by a general equation; (4) both lines are given by an equation with an angular coefficient.
2) Let j be the angle between two straight lines on a plane, and let these straight lines be defined in a Cartesian coordinate system by the equations y = k 1 x + b 1 and y =k 2 x + b 2 .
Then tan j = .
3) Explore the relative position of two straight lines, given by general equations in the Cartesian coordinate system, and fill out the table:
The distance from a point to a straight line on a plane.
Let the straight line l on a plane in the Cartesian coordinate system be given by the general equation Ax + By + C = 0. Let us find the distance from the point M(x 0 , y 0) to the straight line l.
The distance from point M to straight line l is the length of the perpendicular HM (H О l, HM ^ l).
The vector and the normal vector to the line l are collinear, so | | = | | | | and | | = .
Let the coordinates of the point H be (x,y).
Since the point H belongs to the line l, then Ax + By + C = 0 (*).
Coordinates of vectors and: = (x 0 - x, y 0 - y), = (A, B).
| | = 
= 
= 
(C = -Ax - By, see (*))
Theorem. Let the straight line l be specified in the Cartesian coordinate system by the general equation Ax + By + C = 0. Then the distance from the point M(x 0 , y 0) to this straight line is calculated by the formula: r (M; l) = .
Exercises.
1) Derive a formula for calculating the distance from a point to a line if: (1) the line is given parametrically; (2) the line is given to the canonical equations; (3) the straight line is given by an equation with an angular coefficient.
2) Write the equation of a circle tangent to the line 3x – y = 0, with center at point Q(-2,4).
3) Write the equations of the lines dividing the angles formed by the intersection of the lines 2x + y - 1 = 0 and x + y + 1 = 0, in half.
§ 27. Analytical definition of a plane in space
Definition. The normal vector to the plane we will call a non-zero vector, any representative of which is perpendicular to a given plane.
Comment. It is clear that if at least one representative of the vector is perpendicular to the plane, then all other representatives of the vector are perpendicular to this plane.
Let a Cartesian coordinate system be given in space.
Let a plane be given, = (A, B, C) – the normal vector to this plane, point M (x 0 , y 0 , z 0) belongs to plane a.
For any point N(x, y, z) of plane a, the vectors and are orthogonal, that is, their scalar product is equal to zero: = 0. Let us write the last equality in coordinates: A(x - x 0) + B(y - y 0) + C(z - z 0) = 0.
Let -Ax 0 - By 0 - Cz 0 = D, then Ax + By + Cz + D = 0.
Let us take a point K (x, y) such that Ax + By + Cz + D = 0. Since D = -Ax 0 - By 0 - Cz 0, then A(x - x 0) + B(y - y 0) + C(z - z 0) = 0. Since the coordinates of the directed segment = (x - x 0, y - y 0, z - z 0), the last equality means that ^, and, therefore, K О a.
So, we have proven the following theorem:
Theorem. Any plane in space in a Cartesian coordinate system can be specified by an equation of the form Ax + By + Cz + D = 0 (A 2 + B 2 + C 2 ≠ 0), where (A, B, C) are the coordinates of the normal vector to this plane.
The opposite is also true.
Theorem. Any equation of the form Ax + By + Cz + D = 0 (A 2 + B 2 + C 2 ≠ 0) in the Cartesian coordinate system specifies a certain plane, and (A, B, C) are the coordinates of the normal vector to this plane.
Proof.
Take a point M (x 0 , y 0 , z 0) such that Ax 0 + By 0 + Cz 0 + D = 0 and vector = (A, B, C) ( ≠ q).
A plane (and only one) passes through point M perpendicular to the vector. According to the previous theorem, this plane is given by the equation Ax + By + Cz + D = 0.
Definition. An equation of the form Ax + By + Cz + D = 0 (A 2 + B 2 + C 2 ≠ 0) is called general plane equation.
Example.
Let's write the equation of the plane passing through the points M (0,2,4), N (1,-1,0) and K (-1,0,5).
1. Find the coordinates of the normal vector to the plane (MNK). Because vector product´ is orthogonal to non-collinear vectors and , then the vector is collinear ´ .
= (1, -3, -4), = (-1, -2, 1);
´ = 
,
´ = (-11, 3, -5).
So, as the normal vector we take the vector = (-11, 3, -5).
2. Let us now use the results of the first theorem:
equation of this plane A(x - x 0) + B(y - y 0) + C(z - z 0) = 0, where (A, B, C) are the coordinates of the normal vector, (x 0 , y 0 , z 0) – coordinates of a point lying in the plane (for example, point M).
11(x - 0) + 3(y - 2) - 5(z - 4) = 0
11x + 3y – 5z + 14 = 0
Answer: -11x + 3y - 5z + 14 = 0.
Exercises.
1) Write the equation of the plane if
(1) the plane passes through the point M (-2,3,0) parallel to the plane 3x + y + z = 0;
(2) the plane contains the (Ox) axis and is perpendicular to the x + 2y – 5z + 7 = 0 plane.
2) Write the equation of the plane passing through the three given points.
§ 28. Analytical definition of a half-space*
Comment*. Let some plane be fixed. Under half-space we will understand the set of points lying on one side of a given plane, that is, two points lie in the same half-space if the segment connecting them does not intersect the given plane. This plane is called the border of this half-space. The union of this plane and half-space will be called closed half-space.
Let a Cartesian coordinate system be fixed in space.
Theorem. Let the plane a be given by the general equation Ax + By + Cz + D = 0. Then one of the two half-spaces into which the plane a divides the space is given by the inequality Ax + By + Cz + D > 0, and the second half-space is given by the inequality Ax + By + Cz + D< 0.
Proof.
Let us plot the normal vector = (A, B, C) to the plane a from the point M (x 0 , y 0 , z 0) lying on this plane: = , M О a, MN ^ a. The plane divides space into two half-spaces: b 1 and b 2. It is clear that point N belongs to one of these half-spaces. Without loss of generality, we will assume that N О b 1 .
Let us prove that the half-space b 1 is defined by the inequality Ax + By + Cz + D > 0.
1) Take a point K(x,y,z) in the half-space b 1 . Angle Ð NMK is the angle between the vectors and - acute, therefore the scalar product of these vectors is positive: > 0. Let us write this inequality in coordinates: A(x - x 0) + B(y - y 0) + C(z - z 0) > 0, that is, Ax + By + Cy - Ax 0 - By 0 - C z 0 > 0.
Since M О b 1, then Ax 0 + By 0 + C z 0 + D = 0, therefore -Ax 0 - By 0 - C z 0 = D. Therefore, the last inequality can be written as follows: Ax + By + Cz + D > 0.
2) Take a point L(x,y) such that Ax + By + Cz + D > 0.
Let's rewrite the inequality by replacing D with (-Ax 0 - By 0 - C z 0) (since M О b 1, then Ax 0 + By 0 + C z 0 + D = 0): A(x - x 0) + B(y - y 0) + C(z - z 0) > 0.
A vector with coordinates (x - x 0,y - y 0, z - z 0) is a vector, so the expression A(x - x 0) + B(y - y 0) + C(z - z 0) can be understood , as a scalar product of vectors and . Since the scalar product of vectors and is positive, the angle between them is acute and the point L О b 1 .
Similarly, we can prove that the half-space b 2 is given by the inequality Ax + By + Cz + D< 0.
Notes.
1) It is clear that the proof given above does not depend on the choice of point M in the plane a.
2) It is clear that the same half-space can be defined by different inequalities.
The opposite is also true.
Theorem. Any linear inequality of the form Ax + By + Cz + D > 0 (or Ax + By + Cz + D< 0) (A 2 + B 2 + C 2 ≠ 0) задает в пространстве в декартовой системе координат полупространство с границей Ax + By + Cz + D = 0.
Proof.
The equation Ax + By + Cz + D = 0 (A 2 + B 2 + C 2 ≠ 0) in space defines a certain plane a (see § ...). As was proven in the previous theorem, one of the two half-spaces into which the plane divides the space is given by the inequality Ax Ax + By + Cz + D > 0.
Notes.
1) It is clear that a closed half-space can be defined by a non-strict linear inequality, and any non-strict linear inequality in the Cartesian coordinate system defines a closed half-space.
2) Any convex polyhedron can be defined as the intersection of closed half-spaces (the boundaries of which are planes containing the faces of the polyhedron), that is, analytically - by a system of linear non-strict inequalities.
Exercises.
1) Prove the two theorems presented for an arbitrary affine coordinate system.
2) Is the converse true, that any system of non-strict linear inequalities defines a convex polygon?
Exercise.1) Investigate the relative positions of two planes defined by general equations in the Cartesian coordinate system and fill out the table.
Angle between straight lines in space we will call any of the adjacent angles formed by two straight lines drawn through an arbitrary point parallel to the data.
Let two lines be given in space:
Obviously, the angle φ between straight lines can be taken as the angle between their direction vectors and . Since , then using the formula for the cosine of the angle between vectors we get
The conditions of parallelism and perpendicularity of two straight lines are equivalent to the conditions of parallelism and perpendicularity of their direction vectors and:
Two straight parallel if and only if their corresponding coefficients are proportional, i.e. l 1 parallel l 2 if and only if parallel 
.
Two straight perpendicular if and only if the sum of the products of the corresponding coefficients is equal to zero: .
U goal between line and plane
Let it be straight d- not perpendicular to the θ plane;
d′− projection of a line d to the θ plane;
The smallest angle between straight lines d And d′ we will call angle between a straight line and a plane.
Let us denote it as φ=( d,θ)
If d⊥θ, then ( d,θ)=π/2
Oi→j→k→− rectangular coordinate system.
Plane equation:
θ: Ax+By+Cz+D=0
We assume that the straight line is defined by a point and a direction vector: d[M 0,p→]
Vector n→(A,B,C)⊥θ
Then it remains to find out the angle between the vectors n→ and p→, let us denote it as γ=( n→,p→).
If the angle γ<π/2 , то искомый угол φ=π/2−γ .
If the angle is γ>π/2, then the desired angle is φ=γ−π/2
sinφ=sin(2π−γ)=cosγ
sinφ=sin(γ−2π)=−cosγ
Then, angle between straight line and plane can be calculated using the formula:
sinφ=∣cosγ∣=∣ ∣ Ap 1+Bp 2+Cp 3∣ ∣ √A 2+B 2+C 2√p 21+p 22+p 23
Question29. The concept of quadratic form. Sign definiteness of quadratic forms.
Quadratic form j (x 1, x 2, …, x n) n real variables x 1, x 2, …, x n is called a sum of the form 
, (1)
Where a ij – some numbers called coefficients. Without loss of generality, we can assume that a ij = a ji.
The quadratic form is called valid, If a ij
Î GR. Matrix of quadratic form is called a matrix made up of its coefficients. The quadratic form (1) corresponds to the only symmetric matrix 
That is A T = A. Consequently, quadratic form (1) can be written in matrix form j ( X) = x T Ah, Where x T = (X 1 X 2 … x n). (2)
And, conversely, every symmetric matrix (2) corresponds to a unique quadratic form up to the notation of variables.
Rank of quadratic form is called the rank of its matrix. The quadratic form is called non-degenerate, if its matrix is non-singular A. (recall that the matrix A is called non-degenerate if its determinant is not equal to zero). Otherwise, the quadratic form is degenerate.
positive definite(or strictly positive) if
j ( X) > 0 , for anyone X = (X 1 , X 2 , …, x n), except X = (0, 0, …, 0).
Matrix A positive definite quadratic form j ( X) is also called positive definite. Therefore, a positive definite quadratic form corresponds to a unique positive definite matrix and vice versa.
The quadratic form (1) is called negatively defined(or strictly negative) if
j ( X) < 0, для любого X = (X 1 , X 2 , …, x n), except X = (0, 0, …, 0).
Similarly as above, a matrix of negative definite quadratic form is also called negative definite.
Consequently, the positive (negative) definite quadratic form j ( X) reaches the minimum (maximum) value j ( X*) = 0 at X* = (0, 0, …, 0).
Note that most quadratic forms are not sign-definite, that is, they are neither positive nor negative. Such quadratic forms vanish not only at the origin of the coordinate system, but also at other points.
When n> 2, special criteria are required to check the sign of a quadratic form. Let's look at them.
Major minors quadratic form are called minors:
that is, these are minors of the order of 1, 2, ..., n matrices A, located in the upper left corner, the last of them coincides with the determinant of the matrix A.
Positive Definiteness Criterion (Sylvester criterion)
X) = x T Ah was positive definite, it is necessary and sufficient that all major minors of the matrix A were positive, that is: M 1 > 0, M 2 > 0, …, Mn > 0. Negative certainty criterion In order for the quadratic form j ( X) = x T Ah was negative definite, it is necessary and sufficient that its principal minors of even order be positive, and of odd order - negative, i.e.: M 1 < 0, M 2 > 0, M 3 < 0, …, (–1)n
Let straight lines be given in space l And m. Through some point A of space we draw straight lines l 1 || l And m 1 || m(Fig. 138).
Note that point A can be chosen arbitrarily; in particular, it can lie on one of these lines. If straight l And m intersect, then A can be taken as the point of intersection of these lines ( l 1 = l And m 1 = m).
Angle between non-parallel lines l And m is the value of the smallest of adjacent angles formed by intersecting lines l 1 And m 1 (l 1 || l, m 1 || m). The angle between parallel lines is considered equal to zero.
Angle between straight lines l And m denoted by \(\widehat((l;m))\). From the definition it follows that if it is measured in degrees, then 0° < \(\widehat((l;m)) \) < 90°, and if in radians, then 0 < \(\widehat((l;m)) \) < π / 2 .
Task. Given a cube ABCDA 1 B 1 C 1 D 1 (Fig. 139).
Find the angle between straight lines AB and DC 1.
Straight lines AB and DC 1 crossing. Since straight line DC is parallel to straight line AB, the angle between straight lines AB and DC 1, according to definition, is equal to \(\widehat(C_(1)DC)\).
Therefore, \(\widehat((AB;DC_1))\) = 45°.
Direct l And m are called perpendicular, if \(\widehat((l;m)) \) = π / 2. For example, in a cube
Calculation of the angle between straight lines.
The problem of calculating the angle between two straight lines in space is solved in the same way as in a plane. Let us denote by φ the magnitude of the angle between the lines l 1 And l 2, and through ψ - the magnitude of the angle between the direction vectors A And b these straight lines.
Then if
ψ <90° (рис. 206, а), то φ = ψ; если же ψ >90° (Fig. 206.6), then φ = 180° - ψ. Obviously, in both cases the equality cos φ = |cos ψ| is true. According to the formula (the cosine of the angle between non-zero vectors a and b is equal to scalar product of these vectors divided by the product of their lengths) we have
$$ cos\psi = cos\widehat((a; b)) = \frac(a\cdot b)(|a|\cdot |b|) $$
hence,
$$ cos\phi = \frac(|a\cdot b|)(|a|\cdot |b|) $$
Let the lines be given by their canonical equations
$$ \frac(x-x_1)(a_1)=\frac(y-y_1)(a_2)=\frac(z-z_1)(a_3) \;\; And \;\; \frac(x-x_2)(b_1)=\frac(y-y_2)(b_2)=\frac(z-z_2)(b_3) $$
Then the angle φ between the lines is determined using the formula
$$ cos\phi = \frac(|a_(1)b_1+a_(2)b_2+a_(3)b_3|)(\sqrt((a_1)^2+(a_2)^2+(a_3)^2 )\sqrt((b_1)^2+(b_2)^2+(b_3)^2)) (1)$$
If one of the lines (or both) is given by non-canonical equations, then to calculate the angle you need to find the coordinates of the direction vectors of these lines, and then use formula (1).
Task 1. Calculate the angle between lines
$$ \frac(x+3)(-\sqrt2)=\frac(y)(\sqrt2)=\frac(z-7)(-2) \;\;and\;\; \frac(x)(\sqrt3)=\frac(y+1)(\sqrt3)=\frac(z-1)(\sqrt6) $$
Direction vectors of straight lines have coordinates:
a = (-√2 ; √2 ; -2), b = (√3 ; √3 ; √6 ).
Using formula (1) we find
$$ cos\phi = \frac(|-\sqrt6+\sqrt6-2\sqrt6|)(\sqrt(2+2+4)\sqrt(3+3+6))=\frac(2\sqrt6)( 2\sqrt2\cdot 2\sqrt3)=\frac(1)(2) $$
Therefore, the angle between these lines is 60°.
Task 2. Calculate the angle between lines
$$ \begin(cases)3x-12z+7=0\\x+y-3z-1=0\end(cases) and \begin(cases)4x-y+z=0\\y+z+1 =0\end(cases) $$
Behind the guide vector A On the first line we take the vector product of normal vectors n 1 = (3; 0; -12) and n 2 = (1; 1; -3) planes defining this line. Using the formula \(=\begin(vmatrix) i & j & k \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \end(vmatrix) \) we get
$$ a==\begin(vmatrix) i & j & k \\ 3 & 0 & -12 \\ 1 & 1 & -3 \end(vmatrix)=12i-3i+3k $$
Similarly, we find the direction vector of the second straight line:
$$ b=\begin(vmatrix) i & j & k \\ 4 & -1 & 1 \\ 0 & 1 & 1 \end(vmatrix)=-2i-4i+4k $$
But using formula (1) we calculate the cosine of the desired angle:
$$ cos\phi = \frac(|12\cdot (-2)-3(-4)+3\cdot 4|)(\sqrt(12^2+3^2+3^2)\sqrt(2 ^2+4^2+4^2))=0 $$
Therefore, the angle between these lines is 90°.
Task 3. In the triangular pyramid MABC, the edges MA, MB and MC are mutually perpendicular (Fig. 207);
their lengths are respectively 4, 3, 6. Point D is the middle [MA]. Find the angle φ between lines CA and DB.
Let CA and DB be the direction vectors of straight lines CA and DB.
Let's take point M as the origin of coordinates. By the condition of the equation we have A (4; 0; 0), B(0; 0; 3), C(0; 6; 0), D (2; 0; 0). Therefore \(\overrightarrow(CA)\) = (4; - 6;0), \(\overrightarrow(DB)\)= (-2; 0; 3). Let's use formula (1):
$$ cos\phi=\frac(|4\cdot (-2)+(-6)\cdot 0+0\cdot 3|)(\sqrt(16+36+0)\sqrt(4+0+9 )) $$
Using the cosine table, we find that the angle between straight lines CA and DB is approximately 72°.
Definition. If two lines are given y = k 1 x + b 1, y = k 2 x + b 2, then the acute angle between these lines will be defined as
Two lines are parallel if k 1 = k 2. Two lines are perpendicular if k 1 = -1/ k 2.
Theorem. The lines Ax + Bу + C = 0 and A 1 x + B 1 y + C 1 = 0 are parallel when the coefficients A 1 = λA, B 1 = λB are proportional. If also C 1 = λC, then the lines coincide. The coordinates of the point of intersection of two lines are found as a solution to the system of equations of these lines.
Equation of a line passing through this point
Perpendicular to a given line
Definition. A straight line passing through the point M 1 (x 1, y 1) and perpendicular to the straight line y = kx + b is represented by the equation:
Distance from point to line
Theorem. If a point M(x 0, y 0) is given, then the distance to the line Ax + Bу + C = 0 is determined as
.
Proof. Let point M 1 (x 1, y 1) be the base of the perpendicular dropped from point M to a given straight line. Then the distance between points M and M 1:
(1)
The coordinates x 1 and y 1 can be found by solving the system of equations:
The second equation of the system is the equation of the line passing through given point M 0 is perpendicular to a given straight line. If we transform the first equation of the system to the form:
A(x – x 0) + B(y – y 0) + Ax 0 + By 0 + C = 0,
then, solving, we get:
Substituting these expressions into equation (1), we find:
The theorem has been proven.
Example. Determine the angle between the lines: y = -3 x + 7; y = 2 x + 1.
k 1 = -3; k 2 = 2; tgφ = 
; φ= p /4.
Example. Show that the lines 3x – 5y + 7 = 0 and 10x + 6y – 3 = 0 are perpendicular.
Solution. We find: k 1 = 3/5, k 2 = -5/3, k 1* k 2 = -1, therefore, the lines are perpendicular.
Example. Given are the vertices of the triangle A(0; 1), B (6; 5), C (12; -1). Find the equation of the height drawn from vertex C.
Solution. We find the equation of side AB: 
; 4 x = 6 y – 6;
2 x – 3 y + 3 = 0;
The required height equation has the form: Ax + By + C = 0 or y = kx + b. k = . Then y = . Because the height passes through point C, then its coordinates satisfy this equation: 
from where b = 17. Total: . 
Answer: 3 x + 2 y – 34 = 0.
The equation of a line passing through a given point in a given direction. Equation of a line passing through two given points. The angle between two straight lines. The condition of parallelism and perpendicularity of two straight lines. Determining the point of intersection of two lines
1. Equation of a line passing through a given point A(x 1 , y 1) in a given direction, determined by the slope k,
y - y 1 = k(x - x 1). (1)
This equation defines a pencil of lines passing through a point A(x 1 , y 1), which is called the beam center.
2. Equation of a line passing through two points: A(x 1 , y 1) and B(x 2 , y 2), written like this:
The angular coefficient of a straight line passing through two given points is determined by the formula
3. Angle between straight lines A And B is the angle by which the first straight line must be rotated A around the point of intersection of these lines counterclockwise until it coincides with the second line B. If two straight lines are given by equations with a slope
y = k 1 x + B 1 ,
y = k 2 x + B 2 , (4)
then the angle between them is determined by the formula
It should be noted that in the numerator of the fraction, the slope of the first line is subtracted from the slope of the second line.
If the equations of a line are given in general view
A 1 x + B 1 y + C 1 = 0,
A 2 x + B 2 y + C 2 = 0, (6)
the angle between them is determined by the formula
4. Conditions for parallelism of two lines:
a) If the lines are given by equations (4) with an angular coefficient, then the necessary and sufficient condition their parallelism consists in the equality of their angular coefficients:
k 1 = k 2 . (8)
b) For the case when the lines are given by equations in general form (6), a necessary and sufficient condition for their parallelism is that the coefficients for the corresponding current coordinates in their equations are proportional, i.e.
5. Conditions for perpendicularity of two straight lines:
a) In the case when the lines are given by equations (4) with an angular coefficient, a necessary and sufficient condition for their perpendicularity is that their angular coefficients are inverse in magnitude and opposite in sign, i.e.
This condition can also be written in the form
k 1 k 2 = -1. (11)
b) If the equations of lines are given in general form (6), then the condition for their perpendicularity (necessary and sufficient) is to satisfy the equality
A 1 A 2 + B 1 B 2 = 0. (12)
6. The coordinates of the point of intersection of two lines are found by solving the system of equations (6). Lines (6) intersect if and only if
1. Write the equations of lines passing through the point M, one of which is parallel and the other perpendicular to the given line l.
