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Coordinates of a vector

Any two vectors, u,v, not in the same direction form a base.

For any given vector w, you can express w as a linear combination of u and v. w = au + bv. Then, (a, b) are called the coordinates of the vector w in the base B(u, v).

Usually a couple of perperndicular vectors with the same length (=1) is considered. That's an ortonormal base.

Operations with vectors and its coordinates

Given u = (u1, u2) and v = (v1, v2), two vectors and a ∈ R  , then

a·u = (a·u1, a·u2)

u+v = (u1+v1, u2+v2)

Standard basis

The standard basis is {O, i, j}, where O is a point, i ⊥ j and |i| = |j| = 1

Dot product of vectors

Definition of dot product: 

u · v = |u|· |v| · cos (u ^ v)

Given two vectors u = (u1, u2) and v = (v1, v2), the dot product is 

u · v = u1·v1 + u2·v2 

(it can be proved with the help of cosine theorem).

The dot product or scalar product of two vectors is not a vector, but a real number.

Theorem: two vectors u and v are perpendicular if and only if u · v = 0. In math symbols:

u ⊥ v  ⇔  u • v = 0

The dot product allows you to find the angle between two vectors and the projection of one vector onto the other one.

Tip: if u = (u1, u2), then n = (u1, −u2) or m = (−u1, u2) are perpendicular to u.

Analytic geometry procedures

Be it the following frame of reference: {O, i, j}, where O is a point in the 2d plane and (i, j) is an orthonormal basis.

Position vector

In this frame, any point P is identified with a vector OP. The coordinates of P are the coordinates of the vector OP. OP is called the position vector of P.

So, any point P in the plane has a vector univocally associated OP, wher O is the origin of the standard basis. OP is the position vector of P.

Direction of a line, vector defined by two points

Given a line r in that frame, it is defined by a point P and a vector, v, that tells the direction of the line. r(P, v).

Every line has an infinite set of points and directions (parallel). 

Any vector parallel to a line, r, define its direction. Any two points A, B ∈ r define also the direction of r by the vector AB. Using the addition of vectors, OA + AB = OB ⇒ AB = OB − OA.

Three points in the same line

Three points A, B and C are in the same line if AB ∥ BC

Middle point of a line segment

To find out the coordinates of the middle point M (m1, m2) of a line segment AB, where A (a1, a2) and B (b1, b2), you should consider that AM = 1/2 AB ⇒ (m1 − a1, m2 − a2) = (1/2 (b1 − a1), 1/2 · (b2 − a2)) ⇒ m1 = a1 + 1/2 (b1 − a1) and m2 = a2 + 1/2 (b2 − a2) ⇒ 

m1 = (a1 + b1)/2 and m2 = (a2 + b2)/2

Activities

1. If u (–2, 5) y v (1, – 4) are two vectors, work out the coordinates of the following linear combinations: a) 2u+v; b) u–v; c) 3u+(1/3)v; d) –(1/2)u–2 v

2. Given u (3, – 4) y v (–1, 3). Calculate: a) u·v and v·u; b)|u |, | v | and u^v; c) The value of k so that (4, k) is a vector perpendicular to v.

3. Be it ABC the triangle given by its vertices A(3, 7), B(4, 10) and C(0, 0). Prove that it is isosceles.

Equations of a straight line

Line formulas

X is any point of a line r, defined by the point P and the vector v.

1. Vector form: 

OX = OP + λ v

2. Parametric form. Expressing last formula with coordinates, we get: (x, y) = (p1, p2) + λ (v1, v2), then 

x = p1 + λv1

y = p2 + λv2

λ is a parameter. Each value of λ gives you the coordinates (x, y) of a point of r.

3. Cartesian form.

Isolating λ you can get the following expression:

(x - p1)/v1 = (y - p2)/v2

4. General form. Is an expression like the following one, obtained from the last equation:

Ax + By + C = 0

It happen that the vector n (A, B) is perpendicular to the line r. It is called normal vector of the line.

5. Slope-intercept form. If we leave y on one side of that equation, then we get

y = mx + n

There, m is called the slope of the line and n mark its Y-intercept. Given v (v1, v2), the vector of a line r, then the slope of r is

m = v2/v1

Link.

Families of lines

Relative positions between straight lines

Paralell lines and perpendicular lines

Two lines are paralell if their direction vectors are also paralell. On the other hand, two lines are perpendicular if their direction vectors are perpendicular. So this question is a matter of finding direction vectors. 

But, in some cases its easier to proceed in other way. For example, in general form, Ax + By + C = 0, (A, B) is a normal vector. So if you want to know if two lines given by their general equations are paralell or perpendicular, you can proceed by comparing their normal vectors.

Example: 

Given the lines r ≡ 6x - 4y + 1 = 0, and s ≡ 9x - 6y - 7 = 0  ⇒ n(6, -4) and m(9 - 6) are their normal vectors. Then

r ∥ s ⇔ n ∥ m ⇔ 6/9 = -4/-6. So the lines are paralell.

Relative positions between straight lines

Two lines given by its general equation, r:Ax+By+C=0 and r':A'x+B'y+C'=0 can be:

• Paralell lines, if A/A' = B/B' ≠ C/C'

• Intersecting lines, if A/A' ≠ B/B'

• The same line (although expressed differently), if A/A' = B/B' = C/C'

Distances and angles between straight lines

Distance between two points

The distance between two points is de length of the vector they form (see that above).

Distance between a point and a line

d(P,r) = |Ax0+By0+C|/√(A²+B²), where r : Ax + By + C = 0 and P = (x0, y0)

Distance between two lines

Just take one point on one line and apply the last formula.

Angle between two lines

Take a director vector for each line and find out the angle between them as we did above (see the dot product item).

Activities

1. Find the coordinates of the vectors MN and NM, where M   (7, –5) y N (–2, –11).

2. Find out if the following points are aligned P   (7, 11), Q (4, –3) y R (10, 25).

3. Halla las coordenadas del punto A sabiendo que B(–2, 1) y AB = (–5, 6).

4. Determina los valores de los parámetros a y b donde AB = (–3, –2), A(a, 2) y B(–2, b).

5. Find in all its forms the equation of the straight line going through: a) A(-1, -1), B(3,3); b) A(0,4), B(6,0); c) A(3,5), B(-1,5); d) A(3,5), B(3,2).

6. Find two points, P and Q of the line r: 2x-3y+6=0. Check if the vector PQ is perpendicular to v=(2, -3).

7. Find a line parallel and another perpendicular to r:(x+5)/3=(y-1)/-2 going through point (7,-5)

8. Find the line parallel to r:5x-y+4=0 and going through (3, -5) and the line perpendicular to r going to (0,0).

9. Tell the relative possition of the following couples of lines: a) r : 3x + 5y – 8 = 0 and s : 6x + 10y + 4 = 0; b) r : 2x + y – 6 = 0 and s : x – y = 0; c) 

10. Calculate the angle between this two lines: a) r : x + 2y – 17 = 0 and s : 3x – 5y + 4 = 0; b) r : y = 5x – 1 and s : y = 4x + 3

11. Find the distance between point P(2, -1) and line r:(x-1)/3=(2-y)/2

12. Find the distance between the origin and the line (x-3)/-1=(y-2)/2

13. Find the distance between this two lines: r:x/2=(y-1)/5 and s:x/2=(y-3)/5

14. Find the angle between the lines r:(x-2)/3=(y+1)/3 and s:x=1-2t, y=-3+t

15. The points (0, -2), (1, 1), (5, 2) and (4, -1) are the vertices of a quadrilateral. Find the equations of its diagonals and their lengths. What kind of quadrilateral is it?

16. Find the center of a parallelogram in which three of its vertices are known: (5, -1), (9, 5) and (-1, -5). Are there more than one solution? Why? Show it with a drawing?

Answer key: 1. MN = (–2, –11) – (7,–5) = (–9,–6); NM = (7,–5) – (–2,–11) = (9,6); 2. PQ = (–3,–14), QR = (6,28)... 

Exam questions

The exam will provide a real situation in which the student should prove to be able to solve the following questions:

• Find the vector determined by two points.

• Calculate the length of a vector given its coordinates or the distance between two points.

• Draw the addition, subtraction or any linear combination of two vectors.

• Tell if two vectors are parallel, perpendicular or none of them.

• Find the angle between two vectors.

• Find the equation of a line given two of its points in all of its forms.

• Know the meaning of the slope and y-intercept of a line. Find those values on any given line.

It's allowed (and helpful) to use drawing instruments, but the answers must be given by the methods of Analytic Geometry.

Links & Games

Battle of squares.

Addition of vectors (Fourier).

Numbered heads together.

Saberes básicos implicados en la unidad

A. Sentido numérico.

MATE.1.A.1. Sentido de las operaciones.

MATE.1.A.1.1 Adición y producto escalar de vectores: propiedades y representaciones.

MATE.1.A.2. Relaciones.

MATE.1.A.2.2 Conjunto de vectores: estructura, comprensión y propiedades. Módulo de un vector, coordenada de un vector con respecto a una base, ángulo entre dos vectores y proyección ortogonal.

B. Sentido de la medida.

MATE.1.B.1. Medición.

MATE.1.B.1.1 Cálculo de longitudes y medidas angulares: uso de la trigonometría. Razones trigonométricas de un ángulo cualquiera medido en grado s o en radianes. Demostración de las identidades trigonométricas. Razones trigonométricas del ángulo suma, el ángulo diferencia, el ángulo doble y el ángulo mitad. Cálculo de las razones trigonométricas de un ángulo cualquiera empleando las principales fórmulas trigonométricas. Aplicación de las razones trigonométricas, el teorema de los senos y el teorema del coseno en la resolución de triángulos y de problemas geométricos de contexto real. Demostración del teorema del seno y del coseno.

C. Sentido espacial

MATE.1.C.1. Formas geométricas de dos dimensiones.

MATE.1.C.1.1 Objetos geométricos de dos dimensiones: análisis de las propiedades y determinación de sus atributos. Manejo de triángulos, paralelogramos y otras figuras planas.

MATE.1.C.1.2 Resolución de problemas relativos a objetos geométricos en el plano representados con coordenadas cartesianas. Ecuaciones de la recta en el espacio bidimensional. Estudio de la posición relativa de puntos y rectas en el plano. Lugares geométricos: ecuación de la recta mediatriz. Estudio de la simetría en el plano: punto simétrico respecto de otro punto y de una recta; recta simétrica respecto de otra recta. Aplicación de los números complejos para la construcción de polígonos regulares.

MATE.1.C.2. Localización y sistemas de representación.

MATE.1.C.2.1 Relaciones de objetos geométricos en el plano: representación y exploración con ayuda de herramientas digitales.

MATE.1.C.2.2 Expresiones algebraicas de objetos geométricos en el plano: selección de la más adecuada en función de la situación a resolver.

MATE.1.C.3. Visualización, razonamiento y modelización geométrica.

MATE.1.C.3.1 Representación de objetos geométricos en el plano mediante herramientas digitales.

MATE.1.C.3.2 Modelos matemáticos (geométricos, algebraicos, grafos...) en la resolución de problemas en el plano. Conexiones con otras disciplinas y áreas de interés.

MATE.1.C.3.3 Conjeturas geométricas en el plano: validación por medio de la deducción y la demostración de teoremas.

MATE.1.C.3.4 Modelización de la posición y el movimiento de un objeto en el plano mediante vectores.

MATE.1.C.3.5 La geometría en el patrimonio cultural y artístico de Andalucía.

Criterios de evaluación

3. Manejar la operación del producto escalar y sus consecuencias. Entender los conceptos de base ortogonal y ortonormal. Distinguir y manejarse con precisión en el plano euclídeo y en el plano métrico, utilizando en ambos casos sus herramientas y propiedades. CMCT.

Estándares de aprendizaje evaluables

3.1. Emplea con asiduidad las consecuencias de la definición de producto escalar para normalizar vectores, calcular el coseno de un ángulo, estudiar la ortogonalidad de dos vectores o la proyección de un vector sobre otro. 

3.2. Calcula la expresión analítica del producto escalar, del módulo y del coseno del ángulo.