# Unit 8. ANALYTIC GEOMETRY.

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1 Unit 8. ANALYTIC GEOMETRY. 1. VECTORS IN THE PLANE A vector is a line segment running from point A (tail) to point B (head). 1.1 DIRECTION OF A VECTOR The direction of a vector is the direction of the line which contains the vector or any line which is parallel to it. 1.2 COMPONENTS OR COORDINATES OF A VECTOR If the coordinates of A and B are: coordinates of the vector A B are: A B x x, y ) ( y1, then the Solved Examples: Example 1: Find the components of the vector : UNIT 8. Analytic Geometry 1

2 Example 2: The vector has the components (5, 2). Find the coordinates of A if the terminal point is known as B(12, 3). Example 3: Calculate the coordinates of Point D so that points A( 1, 2), B(4, 1), C(5, 2) and D form a parallelogram. 1.3 POSITION VECTOR The vector that joins the origin coordinates, O=(0,0), with a point, P, is the position vector of point P. 1.4 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. UNIT 8. Analytic Geometry 2

3 The magnitude of a vector can be calculated (using the Pythagoras Theorem) if the coordinates of the endpoints are known: Solved examples: Example 1: Calculate the magnitude of the following vector u : Example 2: Calculate the magnitude of the vector determined by the endpoints A and B: Example 3: Calculate the value of k knowing the magnitude of the vector = (k, 3) is UNIT VECTOR The unit vector has a magnitude of one. 1.6 NORMALISING A VECTOR Normalising a vector is obtaining a unit vector in the same direction. To normalise a vector, divide the vector by its magnitude. Examples: If is a vector of components (3, 4), find a unit vector in the same direction. UNIT 8. Analytic Geometry 3

4 2. OPERATION WITH VECTORS 2.1 ADDING VECTORS To add two vectors and graphically, 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. 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 add two vectors, add their components. 2.2 SUBTRACTING VECTORS. To subtract two vectors and, add with the opposite of. 2.3 SCALAR MULTIPLICATION The product of a number, k, by a vector is another vector. The result is another vector in the same direction of if k is positive or in the UNIT 8. Analytic Geometry 4

5 opposite direction of if k is negative. The magnitude of the vector is. Example: 3. CARTESIAN COORDINATES In a system formed by a point, O, and an orthonormal basis at each point, P, there is a corresponding vector,, on the plane such that: The coefficients x and y of the linear combination are called coordinates of point P. The first, x, is the abscissa. The second, y, is the ordinate. UNIT 8. Analytic Geometry 5

6 As the linear combination is unique, each point corresponds to a pair of numbers and a each pair of numbers to a point. 4. DISTANCE BETWEEN TWO POINTS Example: Calculate the distance between points A(2, 1) and B( 3, 2). Solved Problems: 1. Prove that the points: A (1, 7), B (4,6) and C (1, 3) belong to the circumference of a circle whose centre is (1, 2). If O is the centre of the circle, the distances from O to A, B, and C should be equal: UNIT 8. Analytic Geometry 6

7 2. Identify the type of triangle determined by points A (4, 3), B (3, 0) and C (0, 1). It is an isosceles triangle. If: 3. Calculate the value ofa if the distance between points A = (0, a) and B = (1, 2) is equal to 1. UNIT 8. Analytic Geometry 7

8 5. MIDPOINT Solved examples: 1. Calculate the coordinates of the midpoint of 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 M 1 = (2, 1), M 2 = (3, 3) and M 3 = (6, 2) are the midpoints of the sides that make up a triangle, what are the coordinates of the vertices? UNIT 8. Analytic Geometry 8

9 x 1 = 7; x 2 = 7; x 3 = 1 y 1 = 4; y 2 = 0; y 3 = 3 A=(7, 4);B=(5, 0) C=( 1, 2) 6. SLOPE 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: UNIT 8. Analytic Geometry 9

10 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. Solved Examples: Ex 1: The slope of the line that passes through points A = (2, 1) and B = (4, 7) is: Ex 2: The line passes through points A = (1, 2) and B = (1, 7) and has no slope, since division by 0 is undefined. If the angle between the line and the positive x-axis is acute, the slope is positive and grows as the angle increases. If the angle between the line and the positive x-axis is obtuse, the slope is negative and diminishes as the angle increases. UNIT 8. Analytic Geometry 10

11 7. DIFFERENT FORMULAE TO DEFINE A LINE 7.1 VECTOR EQUATION OF A LINE A line is defined as the set of aligned points on the plane with a point, P, and a directional vector. If P(x 1, y 1 ) is a point on the line and the vector direction as, then equals multiplied by a scalar unit: has the same Solved Examples: Ex 1. A line passes through point A = ( 1, 3) and has a directional vector with components (2, 5). Determine the equation of the vector. Ex 2. Write the vector equation of the line which passes through points A = (1, 2) and B = ( 2, 5). UNIT 8. Analytic Geometry 11

12 7.2 PARAMETRIC FORM Solved Examples: Ex 1. A line through point A =( 1, 3) has a direction vector of = (2, 5). Write the equation for this vector in parametric form. Ex 2. Write the equation of the line which passes through points A = (1, 2) and B = ( 2, 5) in parametric form. UNIT 8. Analytic Geometry 12

13 7.3 POINT-SLOPE FORM m is the slope of the line, and (x 1, y 1 ) is any point on the line. Solved Examples: Ex 1. Calculate the point-slope form equation of the line passing through points A = ( 2, 3) and B = (4, 2). Ex 2. Calculate the point-slope form equation of the line with a slope of 45 which passes through point ( 2, 3). Ex 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. 7.4 TWO-POINTS FORM If the values of parameter t in the parametric equations are equal, then: UNIT 8. Analytic Geometry 13

14 The two-point form equation of the line can also be written as: Solved Example: Ex 1: Determine the equation of the line that passes through points A = (1, 2) and B = ( 2, 5). 7.5 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 director vector is: Solved Examples: Ex 1: Determine the equation in general form of the line that passes through point A = (1, 5) and has a slope of m = 2. Ex 2: Write the equation in general form of the line that passes through points A = (1, 2) and B = ( 2, 5). UNIT 8. Analytic Geometry 14

15 7.6 SLOPE-INTERCEPT FORM If the value of y in the general form equation is isolated, the slope intercept form of the line is obtained: The coefficient of x is the slope, which is denoted as m. The independent term is the y-intercept, which is denoted as b. Example: Calculate the equation (in slope intercept form) of the line that passes through point A = (1,5) and has a slope m = 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. UNIT 8. Analytic Geometry 15

16 Two lines are parallel if the angle between them is 0º. Solved Examples: Ex 1: Calculate k so that the lines r x+2y-3 = 0 and s x ky+4 = 0, are parallel. Ex 2: Determine the equation for the line parallel to r x+2y+3 = 0 that passes through point A = (3, 5). Ex 3: Determine the equation for the line parallel to r 3x+2y 4 = 0 that passes through point A = (2, 3) k = 0 k = 12 3x + 2y 12= 0 Ex 4: The line r 3x+ny 7 = 0 passes through point A = (3, 2), and it is parallel to the line s mx+2y 13 = 0. Calculate the values of m and n. UNIT 8. Analytic Geometry 16

17 9. 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. Solved Examples: Ex 1: Determine the equation of the line that is perpendicular to r x+2y+3=0 and passes through point A=(3, 5). Ex 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 line t 5x 8y + 12 = 0. Ex 3: Calculate k so that lines r x + 2y 3 = 0 and s x ky + 4 = 0 are perpendicular. UNIT 8. Analytic Geometry 17

18 10. ANGLE BETWEEN TWO LINES The angle between two lines is the smaller of the angles formed by th e intersection of the two lines. The angle can be obtained from: 1. Their slopes: 2. Their director vectors: Solved Examples: Ex 1: Find the angle between lines r and s, if their directional vectors are: = ( 2, 1) and = (2, 3). Ex 2: Given the lines r 3x + y - 1 = 0 and s 2x + my - 8 = 0, calculate the value of m so that both lines form an angle of 45. UNIT 8. Analytic Geometry 18

19 11. DISTANCE FROM A POINT TO A LINE The distance from a point to a straight line is the length of a line segment drawn from the point that forms a perpendicular angle with the straight line. Solved Examples: Ex 1: Find the distance from Point P = (2, 1) to the line r 3 x + 4 y = 0. Ex 2: Calculate the distance between the following lines: UNIT 8. Analytic Geometry 19

20 EXERCISES 1. Write the equation (in all possible forms) of the line that passes through points A = (1, 2) and B = (2, 5). 2. Identify the type of triangle formed by these points: A = (6, 0), B = (3, 0) and C = (6, 3). 3. Determine the slope and y-intercept of this line: 3x + 2y 7 = Find the equation of the line r which passes through point A = (1, 5) and is parallel to line s 2x + y + 2 = Find the equation of the line that passes through point (2, 3) and is parallel to the straight line that joins points (4, 1) and ( 2, 2). 6. Points A = ( 1, 3) and B = (3, 3) are vertices of an isosceles triangle ABC that has its apex C on line 2x 4y + 3 = 0. If AC and BC are the equal sides, calculate the coordinates of point C. 7. Line r 3x + ny 7 = 0 passes through point A = (3, 2) and is parallel to line s mx + 2y 13 = 0. Calculate the values of m and n. 8. Given a triangle ABC with coordinates A = (0, 0), B = (4, 0) and C = (4, 4), calculate the equation of the angle bisector that passes through the vertex, C. 9. A parallelogram has a vertex A = (8, 0), and the point of intersection of its two diagonals is M = (6, 2). If the other vertex is at the origin, calculate: a) The other two vertices. b) The equations of the diagonals. c) The length of the diagonal. 10. Identify the type of triangle formed by points: A = (4, 3), B = (3, 0) and C = (0, 1). 11. Calculate the equation of the line that passes through point P = ( 3, 2) and is perpendicular to line r 8x y 1 = Line r x + 2y 9 = 0 is the perpendicular bisector of the line segment AB whose endpoint A has the coordinates (2, 1). Find the coordinates of the other endpoint. 13. Calculate the angle between the lines whose equations are: UNIT 8. Analytic Geometry 20

21 14. A straight line is parallel to line r 5x + 8y 12 = 0, and it is 6 units from the origin. What is the equation of this line? 15. Determine the equations of the angle bisectors formed by these lines: 16. The vertices of a parallelogram are A = (3, 0), B = (1, 4), C = ( 3, 2) and D = ( 1, 2). Calculate its area. 17. Given the triangle formed by points A = ( 1, 1), B = (7, 5) and C = (2, 7), calculate the equations of its heights and determine the orthocenter of the triangle. 18. A line is perpendicular to line r 5x 7y + 12 = 0, and it is 4 units away from the origin. Determine the equation of this line. UNIT 8. Analytic Geometry 21

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