unit 7 analytic geometry - wordpress.com · 5/7/2017 · unit 7 analytic geometry 3. scalar...

UNIT 7 ANALYTIC GEOMETRY

UNIT 7 ANALYTIC GEOMETRYIt is impossible to be a mathematician without being a poet in soul. Sofia Kovalevskaya

1. VECTORS IN THE PLANE

A vector is a line segment running from point A (tail) to point B (head).

Each vector has a magnitude (also referred to as length) and a direction.

Direction of a Vector

This is the direction of the line which contains the vector or any line which is parallel to it.

Magnitude of a Vector

The magnitude of the vector is the length of the line

segment . It is denoted by .

The magnitude of a vector is always a positive number or zero.

1/21 Mercedes López Muñoz IES Universidad Laboral


Components or Coordinates of a Vector

If the coordinates of A and B are:

Examples

1. Find the components of the vector :

2. The vector has the components (5, −2). Find the coordinates of A if the terminal point is known as B(12, −3).

Magnitude of a vector

The magnitude of a vector can be calculated if the coordinates of the endpoints are known:

Given , then .



Position vector

The vector that starts at the point O(0,0) and ends at the point A(x,y) is the position vector of point A.OA =(x,y)-(0,0) (x,y)The components of the vector position of a point are the same as the coordinates of the point.

Equivalent vectors

Two vectors are equivalent if they have the same magnitude and direction.

Equivalent vectors have the same coordinates:

u=(-1,3)-(2.4)=(-3,-1)OR=(-3,-1)-(0.0)=(-3,-1)w=(1,-2)-(4,-1)=(-3,-1)

Free vectors

A free vector is an infinite set of parallel directed line segments. If we simply specify magnitude and direction then any two vectors of the same length and parallel to each other are considered to be identical. We usually represent a free vector by letter .

EXERCISES

1 Calculate the head of the vector knowing that its components are (3, −1) and its tail is A = (−2, 4).

2 Given points A = (0, a) and B = (1, 2), calculate the value of a if the magnitude of the vector is one.

(Solutions at the end)



2. OPERATIONS WITH VECTORS.

Addition

To add two vectors, and , we can use two different procedures:

a) join the tail of one with the head of the other vector.

The vector sum equals the distance from the tail of the first vector to the head of the second vector.

b) Parallelogram Rule :If there are two vectors with a common origin and parallel lines to the vectors are drawn, a parallelogram is obtained whose diagonal coincides with the sum of the vectors.

To find the coordinates of the vector sum, add the components of the two vectors

Multiplication

The product of a number, k, by a vector is another vector:

In the same direction of if k is positive.

In the opposite direction of if k is negative.

Of magnitude



Example

Linear Combination

Given the numbers a1, a2, ..., an and the vectors v1, v2, ..., vn, a linear combination is

each of the vectors of the form:

Examples:

1. Given the vectors , calculate the linear combination

vector

2. Can the vector be expressed as a linear combination of the vectors



?

Mid point of a segment

The coordinates of the mid point M of a segment are a linear

combination of the positions vectors of the endpoints.

Examples

1. Calculate the coordinates of the midpoint from the line segment AB.

2. Calculate the coordinates of Point C in the line segment AC, knowing that the midpoint is B = (2, −2) and an endpoint is A = (−3, 1).



3.If M1 = (2, 1), M2 = (3, 3) and M3 = (6, 2) are the midpoints of the sides that make up a

triangle, what are the coordinates of the vertices?

x1 = 7 x5 = 7 x3 = −1

y1 = 4 y5 = 0 y3 = 3

A(7, 4)B(5, 0) C(−1, 2)

4. If the line segment AB with endpoints A = (1,3) and B = (7, 5) is divided into four equalparts, what are the coordinates of the points of division?



3. SCALAR PRODUCT OF TWO VECTORS

The scalar product or dot product of two vectors and is equal to:

Example

It can also be expressed as:

Example

Scalar Projection

From a geometyrical point of vue, the scalar product of two vectors is the product of themodule of one of them by the projection of the other vector on it.



Angle between two vectors

Using the two formulas for the scalar product:

and ,

the angle between the vectors and is given by the formula:

Example:

Exercises:

3. Calculate the dot product and the angle formed by the following vectors:

a) = (3, 4) and = (−8, 6)

b) = (5, 6) and = (−1, 4)

c) = (3, 5) and = (−1, 6)

4. Given the vectors = (2, k) and = (3, −2), calculate the value of k so that the

vectors and are:

a) Perpendicular.

b) Parallel.

c) Make an angle of 60°.

5. Find the value of k if the angle between = (3, k) and = (2, −1) is:

a) 90°

b) 0º

c) 45º

6. Calculate the angles of the triangle with vertices: A = (6,0), B = (3,5) and C = (−1,−1).



(Solutions at the end)

Orthogonal Vectors

Two vectors are orthogonal or perpendicular if their scalar product is zero.

Example

Not perpendicular.

Exercises ,more exercises and more exercises

4. EQUATIONS OF THE LINE

Vector equation of a line

A line is defined as the set of alligned points on the plane with a point, P, and a

directional vector, .


http://www.vitutor.com/geo/vec/aActivdades.html

http://www.vitutor.com/geometry/vec/vector_problems.html

http://www.vitutor.com/geo/vec/a_a.html


If P(x1, y1) is a point on the line and the vector has the same direction as , then

equals multiplied by a scalar unit:

Examples

1. A line passes through point A = (−1, 3) and has a directional vector with components(2, 5). Determine the equation of the vector.

2. Write the vector equation of the line which passes through the points A = (1, 2) and B = (−2, 5).

Parametric equations of the line



Examples

1. A line through point A = (−1, 3) has a direction vector of = (2, 5). Write the equationfor this vector in parametric form.

2. In parametric form, write the equation of the line which passes through the points A =(1, 2) and B = (−2, 5).

Continuous form

If the value of the parameter t from the parametric equations are equal, then:

General form

A, B and C are constants and the values of A and B cannot both be equal to zero.

The equation is usually written with a positive value for A.

The slope of the line is:

The direction vector is:

The normal vector is n⃗=(A,B) and is perpendicular to the direction vector (as you can check multiplying them)


http://www.vitutor.com/geometry/line/parametric_form.html


Slope-intercept form

If the value of y in the general form equation is isolated, the slope–intercept form of theline is obtained:

The coefficient of x is them slope, which is denoted as m.

The independent term is the y-intercept which is denoted as b. (In Spain and other countries, we use letter n).

Example

Calculate the equation (in slope–intercept form) of the line that passes through Point A =(1,5) and has a slope m = −2.

y=mx+ny=−2x+n5=−2·1+n

n=7

Point-slope form

m is the slope of the line and (x1, y1) is any point on the line.

Examples

1. Calculate the point-slope form equation of the line passing through points A = (−2, −3)and B = (4, 2).



2. Calculate the equation of the line with a slope of 45° which passes through the point (−2, −3).

3. A line passes through point A = (−1, 3) and has a direction vector of = (2, 5). Write the equation of the line in point-slope form.

4. SLOPE. INCIDENCE OF LINES

The slope is the inclination of a line with respect to the x-axis.

It is denoted by the letter m.

Slope given two points:

Slope given the angle:

Slope given a vector of the line:

Slope given the equation of the line:

Two lines are parallel if their slopes are equal.

Two lines are perpendicular if their slopes are the inverse of each other and their signs are opposite.



PARALLEL LINES

Two lines are parallel if their slopes are equal.

Two lines are parallel if the respective coefficients of x and y are proportional.

Two lines are parallel if their directional vectors are equal.

Two lines are parallel if the angle between them is 0º.

Examples

1. Calculate k so that the lines r x + 2y − 3 = 0 and s x − ky + 4 = 0, are parallel.≡ ≡

2. Determine the equation for the line parallel to r x + 2 y + 3 = 0 that passes through ≡the point A = (3, 5).



3. Determine the equation for the line parallel to r 3x + 2y − 4 = 0 that passes through ≡the point A = (2, 3).

3 · 2 + 2· 3 + k = 0 k = −12

3x + 2y − 12= 0

4. The line r 3x + ny − 7 = 0 passes through the point A = (3, 2) and is parallel to the ≡line s mx + 2y − 13 = 0. Calculate the values of m and n.≡

PERPENDICULAR LINES

If two lines are perpendicular, their slopes are the inverse of each other and their signs are opposite.

Two lines are perpendicular if their directional vectors are perpendicular.



Examples

1. Determine the equation of the line that is perpendicular to r x + 2 y + 3 = 0 and ≡passes through the point A = (3, 5).

2. Given the lines r 3x + 5y − 13 = 0 and s 4x − 3y + 2 = 0, calculate the equation of ≡ ≡the line that passes through their point of intersection and is perpendicular to the line t 5x − 8y + 12 = 0≡

3. Calculate k so that the lines r x + 2y − 3 = 0 and s x − ky + 4 = 0 are perpendicular.≡ ≡

INCIDENCE OF LINES IN GENERAL FORM:

Given the lines in general form Ax+By+C=0 and A'x+B'y+C'=0, the lines are:

. INTERSECTING if m≠m' , that is, if AA '

≠BB '

.

. PARALLEL if m=m' , that is, if AA '

=BB '

≠CC '

.

. COINCIDENT if m=m' and n=n ' , that is, if AA '

=BB '

=CC '



Given the line of equation Ax+By+C=0 , all the lines PARALLEL to it have the equation Ax+By+k=0 , where k is any real number.

Given the line of equation Ax+By+C=0 , all the lines PERPENDICULAR to it have the equation Bx-Ay+k=0 , where k is any real number.

Exercises

More exercises

And even more exercises

SOLUTIONS

1. Calculate the head of the vector knowing that its components are (3, −1) and its tail is A = (−2, 4).

3 = xB − (−2) xB = 1

−1 = yB − 4 yB = 3

B(1, 3)

2. Given points A = (0, a) and B = (1, 2), calculate the value of a if the magnitude of the

vector is one.

3. Calculate the scalar product and the angle formed by the following vectors:

a) = (3, 4) and = (−8, 6)

· = 3 · (−8) + 4 · 6 = 0

b) = (5, 6) and = (−1, 4)

· = 5 · (−1) + 6 · 4 = 19


http://www.vitutor.com/geo/rec/recta_eso.html

http://www.vitutor.com/geometry/line/solved_lines.html

http://www.vitutor.com/geometry/line/line_problems.html


c) = (3, 5) and = (−1, 6)

· = 3 · (−1) + 5 · 6 = 27

4. Given the vectors = (2, k) and = (3, −2), calculate the value of k so that the

vectors and are:

a) Perpendicular.

b) Parallel.

c) Make an angle of 60°.

5. Find the value of k if the angle between = (3, k) and = (2, −1) is:

a) 90°

b) 0°



c) 45°

6. Calculate the angles of the triangle with vertices: A = (6,0), B = (3,5) and C = (−1,−1).


unit 7 analytic geometry - wordpress.com · 5/7/2017 · unit 7 analytic geometry 3. scalar...

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