A wall clock has a minute hand with a length of 0.49 m and an hour hand with a length of 0.26 m. Take the center of the clock as the origin, and use a Cartesian coordinate system with the positive x axis pointing to 3 o'clock and the positive y axis pointing to 12 o'clock. Express the position vectors of the tip of the hour hand (vector A) and the tip of the minute hand (vector B) at the following times.
(a) 12:00
(b) 3:00
(c) 6:00
(d) 9:00
x = r cosθ
y = r sinθ
for the minute hand, after t minutes, θ = -2π * t/60 = -π/30 t
similarly for the hour hand, but it moves 12 times more slowly.
the - sign is because we are measuring the angles clockwise, rather than the usual counter-clockwise.
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 use trigonometry. Let's calculate them step-by-step for each given time:
(a) 12:00:
At 12:00, both the hour and minute hand point directly upward, along the positive y-axis.
So, the position vector of the tip of the hour hand (vector A) will be (0, 0.26) and the position vector of the tip of the minute hand (vector B) will be (0, 0.49).
(b) 3:00:
At 3:00, the hour hand points along the positive x-axis and the minute hand points upwards.
So, the position vector of the tip of the hour hand (vector A) will be (0.26, 0) and the position vector of the tip of the minute hand (vector B) will be (0, 0.49).
(c) 6:00:
At 6:00, both the hour and minute hand point directly downward, along the negative y-axis.
So, the position vector of the tip of the hour hand (vector A) will be (0, -0.26) and the position vector of the tip of the minute hand (vector B) will be (0, 0.49).
(d) 9:00:
At 9:00, the hour hand points along the negative x-axis and the minute hand points downwards.
So, the position vector of the tip of the hour hand (vector A) will be (-0.26, 0) and the position vector of the tip of the minute hand (vector B) will be (0, -0.49).
These are the position vectors for the given times.
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.