Triple Product in Coordinates. Given three vectors , , and , let us express the triple product of these vectors in terms of their coordinates. Consider the triple product

.

The vector product equals

.

Taking its inner product with , we obtain

;

this is a third – order determinant expanded along the last line, i.e.,

.

Thus, the triple product of three vectors equals the third – order determinant of the composed of the coordinates of these vectors.

_{LECTURE 3}

The simplest problem of analytic geometry. The equation of the line in the plane. The curves of the second order. Canonical equation of second order curves. Economic interpretation of the curves of the second order

1. The simplest problem of analytic geometry

C

a₂

onsider the Cartesian rectangular coordinate system in the   plane. Taking the projection of any point М₁ on the x and y  coordinate we obtain two numbers x=a₁ and y=b₁. Take two numbers a₂ and b₂ plot a₂ on the x-axis and b₂ on the y-axis . Having drawn two straight lines parallel to the coordinate axes through these points we find obtain a point M₂ in their intersection.

y

M₂(a₂;b₂)

M₁(a₁;b₁)

b₂

x

0

a₁

Thus, there is a one–to–one correspondence between points in the plane and pairs of numbers.

Analytic geometry (all of its statements, theorems, and formulas) can be constructed on the basis elementary school mathematics. But we use tools of vector algebra in derivations and proofs.

The distance between two points. Let us find the distance between two points М₁ and М₂ in the plane.

y

d M₂(x_2;,y₂)

M₁(x₁;y₁)

0 x

Compose the vector .

The length of this vector is defined by

.

This is the distance between the two given points.

Division of an interval in a given ratio. Suppose given an interval м1м2. Let us find the coordinates a of point м on the interval for which .

Compose the vectors and .

y

M₂(x₂,y₂)

M(x,y)

M₁(x₁,y₁)

0 x

It is known from vector algebra that the condition for two vectors to be collinear is the proportionality of their respective coordinates:

.

From the first fraction we obtain

; ; .

This gives the x coordinate; y is found in a similar way:

; .

To obtain a formula for the midpoint of the interval, we take =1:

; .

2. Straight Lines in the Plane

The equation of a straight line with a slope.

Given a straight line, we denote the angle between this line and the x–axis by  and the interval cut out by the line on the x-axis by b.

у М(х;у)

y-b

B 

b N

0 x

Definition 1. The slope tangent of the angle between a straight line and the x–axis is called the slope of the line and denoted by

k=tan.

Suppose that k is the slope of a line and b is its y–intercept. Let us write an equation of this line.

Take a point M(x;y) and consider the triangle BMN , where for any point М of the straight line under consideration. We obtain

, where and , whence .

Thus, the equation of a straight line with a slope has the form

.

The equation of a straight line with given slope passing through a given point. Suppose that a straight line passes through a point М₀(x₀,y₀) and has slope k.

By analogy with the equation of a straight line with a slope consider the triangle М₀MN; we have for any point М on the under

consideration or .

Thus, the required equation is

y – y₀ = k(x–x₀).

The equation of a straight line passing through two points. Suppose that a straight line passes through two points М₁(х₁;у₁) and М₂(х₂;у₂).

Take a point M(x,y) on the line and consider the vectors

and .

These two vectors и lie on the same straight line and are collinear.

The collinearity condition is the proportionality of the perspective coordinates, i.e.,

This is the equation of a straight line passing through the two given points.

The general equation of a straight line and its analysis.

Definition 2. A first–order equation in variables x and y determines a straight line in the plane.

The general equation has the form ,

where А and В are called the coefficients of the variables.

1. If the free term is С=0, then the equation has the form .

Since х=0 and y=0 satisfy this equation, it follows that the straight line passes through the origin.

2. If the coefficient of х is А=0, then the equation has the form

or , i.e., the line is parallel to the x–axis.

3. If the coefficient of y is B=0, then the equation has the form

,

and the line is parallel to the y–axis.

4. If А=С=0, then the line В у=0, (or у=0) coincides with the x–axis.

5. If В=С=0, then the line А х=0, (or х=0) coincides with the y–axis.

The angle between two straight lines. Parallel and perpendicular lines. Suppose given two straight lines with slopes and . It is seen from the figure that the angle between the lines equals =₂–₁. Using the formula for the tangent of the difference between two angels, we obtain .

Replacing the tangents by the slopes k₁ and k_2, we obtain the following formula for the tangent of the angle between two straight lines:

. (1)

(*) Suppose that the right lines are parallel, i.e., the angle between them is ; substituting it into formula (1), we obtain

.

This fraction vanishes, if k₂–k₁=0.

Thus, two straight lines are parallel if and only if their slopes are equal:

k₂=k₁.

(**) Suppose that two straight lines are perpendicular; then the angle between them is . Substituting it in (1), we obtain

.

This fraction equals infinity when the denominator vanishes:

.

Consequently, the condition for two straight lines to be perpendicular is

.

The mutual arrangement of two straight lines. Given equations of two straight lines

and .

Determine conditions on the coefficient, for these right lines to intersect, be parallel, or coincide.

1. To determine the mutual arrangement of lines, we must analyze the system of equations

If the lines intersect, then this system has a unique solution, and its principal determinant is nonzero:

; ;

.

Thus, if the straight lines intersect, then the coefficients must not be proportional.

2. Suppose that the straight lines are parallel, i.e., they have no common points, and the system of equations has no solution; then the principal determinant vanishes, and the auxiliary determinants are nonzero:

; ;

;

; ; ; ;

.

This is the parallelis condition.

3. When the straight lines coincide, i.e., have many common points, the system of equations has infinitely many solutions. In this case, the auxiliary and principal determinants are zero:

; ;

.

This is the condition for straight lines to coincide.

Distance d from the point M₀(x₀,y₀) to the line is the length of the perpendicular from this point on the line.

The distance d is defined by:

.