To find the position vectors of the tip of the hour hand (vector A) and the tip of the minute hand (vector B) at different times, we can break down the problem into two parts: finding the angles made by the hands with the positive x-axis and then converting those angles into position vectors.
First, let's find the angles made by the hour and minute hands with the positive x-axis for each given time:
(a) 12:00: At 12:00, the minute hand points directly at the positive y-axis (12 o'clock position), while the hour hand points towards the negative x-axis.
Angle made by the hour hand (θ1): Since the hour hand is pointing towards the negative x-axis, the angle made will be 180 degrees or π radians.
Angle made by the minute hand (θ2): Since the minute hand is pointing towards the positive y-axis, the angle made will be 90 degrees or π/2 radians.
(b) 3:00: At 3:00, both the hour and minute hand point along the positive x-axis.
Angle made by the hour hand (θ1): Since the hour hand is pointing along the positive x-axis, the angle made will be 0 degrees or 0 radians.
Angle made by the minute hand (θ2): Since the minute hand is also pointing along the positive x-axis, the angle made will be 0 degrees or 0 radians.
(c) 6:00: At 6:00, the hour hand points towards the positive y-axis (12 o'clock position), while the minute hand points towards the negative x-axis.
Angle made by the hour hand (θ1): Since the hour hand is pointing towards the positive y-axis, the angle made will be 90 degrees or π/2 radians.
Angle made by the minute hand (θ2): Since the minute hand is pointing towards the negative x-axis, the angle made will be 180 degrees or π radians.
(d) 9:00: At 9:00, both the hour and minute hands point along the negative x-axis.
Angle made by the hour hand (θ1): Since the hour hand is pointing along the negative x-axis, the angle made will be 180 degrees or π radians.
Angle made by the minute hand (θ2): Since the minute hand is also pointing along the negative x-axis, the angle made will be 180 degrees or π radians.
Now that we have the angles, we can convert them into position vectors considering the lengths of the hour and minute hands.
For vector A (tip of the hour hand):
(a) 12:00: The position vector of A is (-0.26, 0).
(b) 3:00: The position vector of A is (0.26, 0).
(c) 6:00: The position vector of A is (0, 0.26).
(d) 9:00: The position vector of A is (-0.26, 0).
For vector B (tip of the minute hand):
(a) 12:00: The position vector of B is (0, 0.49).
(b) 3:00: The position vector of B is (0.49, 0).
(c) 6:00: The position vector of B is (0, -0.49).
(d) 9:00: The position vector of B is (-0.49, 0).
Please note that the position vectors are in the form (x, y) where x represents the horizontal position and y represents the vertical position in the Cartesian coordinate system.