At 12:15 PM the direction of P x Q will be -
Understand the Problem
The question is asking about the direction of the cross product of the hour and minute hands of a clock at 12:15 PM. We need to determine the orientation of the resultant vector based on the positions of the hands at that time.
Answer
Toward south
Answer for screen readers
The direction of ( P \times Q ) will be - Toward south.
Steps to Solve
- Identify the positions of the clock hands at 12:15 PM
At 12:15 PM, the hour hand (P) points slightly past the 12 (specifically, at the 1 on the clock) and the minute hand (Q) points directly at the 3.
- Determine the vectors for the hour and minute hands
- The hour hand (P) at 1/4 of the way between 12 and 1 can be represented approximately as: $$ P = (0.5, 0.5) $$
- The minute hand (Q) pointing at the 3 can be represented as: $$ Q = (1, 0) $$
- Perform the cross product of vectors P and Q
Using the cross product formula for vectors in a 3D space (the z-component is of concern here since we are in a 2D plane), we can express it as: $$ P \times Q = P_x Q_y - P_y Q_x $$ Given our vectors:
- ( P_x = 0.5 ), ( P_y = 0.5 )
- ( Q_x = 1 ), ( Q_y = 0 )
Substituting values: $$ P \times Q = (0.5)(0) - (0.5)(1) = 0 - 0.5 = -0.5 $$
- Determine the direction of the resultant vector
In a standard 3D right-hand system, if the calculated cross product is negative, the direction of the vector points downwards (south). Given that the hour arm (P) is above (12 o'clock) and the minute arm (Q) is right (3 o'clock), the resultant vector will point south.
The direction of ( P \times Q ) will be - Toward south.
More Information
The cross product in this context gives us an indication of the perpendicular direction to the plane formed by the two clock hands. By applying the right-hand rule, we can visualize the direction as being downward into the page, which relates to south when aligned with conventional compass directions.
Tips
- Misinterpreting the clock positions: Ensure that the positions of the hands are accurately considered without neglecting the slight angle of the hour hand at 12:15 PM.
- Incorrect cross product calculation: Be careful with signs and factors when performing the calculations for the cross product.
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