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At a function of re-opening Mandela National Stadium,6 people had a simple game in which player number 1 kicked the ball from the origin O to player A by a displacement.(2 6), then player A displaced it to B by (2 -3) , player B to C by (1 0), C to player 1 by (-5 -3) player decided to make a straight kick to the referee who was in position (8,8).

By coincidence, all the people in the game looked in the mirror which was fixed on the x-axis wall and saw their images in different positions.
Task,
(a) (i) Which role does number 1 play in the game?
(ii)By use of vector geometry summarize the touches the players made in a Cartesian plane.
(iii) Which translation vector describes vector AC and AR?
(b) Reduce the relation in the form (X,Y) of the straight path between number 1 and the referee. Hence state the range of the relation.
(c) Determine the position vectors of each people in the mirror.
(d) With clear reasons, justify the properties that all the 6 people were in the mirror.
(20 scores)

1 answer

Let's break down the task step by step:

(a)

(i) Role of Player Number 1

Player number 1 is the initial player who starts the game by kicking the ball from the origin (O) and initiating the series of passes.

(ii) Summary of the Touches Using Vector Geometry

Let's track the displacement of the ball step-by-step on the Cartesian plane:

  1. Player 1 to A: \[ \text{Position of A} = O + (2, 6) = (2, 6) \]

  2. Player A to B: \[ \text{Position of B} = A + (2, -3) = (2, 6) + (2, -3) = (4, 3) \]

  3. Player B to C: \[ \text{Position of C} = B + (1, 0) = (4, 3) + (1, 0) = (5, 3) \]

  4. Player C to Player 1: \[ \text{Position of Player 1} = C + (-5, -3) = (5, 3) + (-5, -3) = (0, 0) \quad \text{(back to origin)} \]

  5. Player 1 to referee (R): \[ \text{Position of R} = (8, 8) \]

Summary of touches:

  • Player 1 to A: (2, 6)
  • Player A to B: (4, 3)
  • Player B to C: (5, 3)
  • Player C to Player 1: (0, 0) (back to origin)
  • Player 1 to Referee: (8, 8)

(iii) Translation Vectors for AC and AR

  • Vector AC: \[ AC = C - A = (5, 3) - (2, 6) = (3, -3) \]

  • Vector AR: \[ AR = R - A = (8, 8) - (2, 6) = (6, 2) \]

(b) Straight Path Between Player 1 and the Referee

To find the equation of the line from point (0, 0) (Player 1) to (8, 8) (Referee), we can use the slope-intercept form:

The slope \( m \) from Player 1 to the referee is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 0}{8 - 0} = 1 \] Thus, the line can be expressed as: \[ y = x \] For \( x \) and \( y \) ranging from 0 to 8, the relation can be expressed as: \[ (x, y) \text{ where } y = x \text{ and } 0 \leq x \leq 8 \]

Range of the relation: The range of \( y \) is also \( [0, 8] \).

(c) Position Vectors of Each Player in the Mirror

In the mirror located on the x-axis, the y-coordinates get reflected. Therefore, the position vectors of each person will have their y-coordinates negated:

  • Player 1: (0, 0) → (0, 0)
  • Player A at (2, 6) → (2, -6)
  • Player B at (4, 3) → (4, -3)
  • Player C at (5, 3) → (5, -3)
  • Referee at (8, 8) → (8, -8)

So, the position vectors in the mirror are:

  • (0, 0)
  • (2, -6)
  • (4, -3)
  • (5, -3)
  • (8, -8)

(d) Justification for the Properties in the Mirror

  • Reflection: The position of each person in front of the mirror results in their reflection, meaning their y-coordinate is inverted while the x-coordinate remains the same. This conforms to the properties of reflection in geometry.
  • Distance Preservation: The distances from each person to the mirror (the x-axis) remain unchanged, thus obeying the fundamental property of reflection where the distance to the reflecting surface remains constant.
  • Orientation: The lineup of the coordinates maintains the same orientation with respect to the x-axis; only the 'upwards' position is mirrored to 'downwards'.

Through these properties, the dynamics of reflection are consistent with how people perceive their own images in a mirror located along the x-axis.