Zainab is standing facing a particular direction initially. Then she turns 90° clockwise, after that she turns 180° anticlockwise. Then she turns 135° anticlockwise. She finally turns 180° clockwise. Now she is facing the north-east direction. In which direction was Zainab facing initially before taking any turns?
East
Understanding Zainab's Directional Changes
This problem requires us to trace Zainab's movements based on a series of directional turns (clockwise and anticlockwise) and determine her initial facing direction. We are given her final direction (North-East) after these turns.
Key Concepts: Directions and Turns
We can visualize directions using a compass rose. Let's define the standard directions and their corresponding angles, assuming North is 0°:
- North (N): 0°
- East (E): 90°
- South (S): 180°
- West (W): 270°
- North-East (NE): 45°
- South-East (SE): 135°
- South-West (SW): 225°
- North-West (NW): 315°
Clockwise (CW) turns increase the angle (turn right).
Anticlockwise (ACW) turns decrease the angle (turn left).
A full circle rotation is 360°.
Step-by-Step Analysis of Zainab's Turns
Let $D_{initial}$ be the direction Zainab was facing initially. We can track the changes:
- Start: Zainab faces $D_{initial}$.
- Turn 1: She turns 90° clockwise. Her new direction is $D_{initial} + 90°$.
- Turn 2: She turns 180° anticlockwise. Her direction becomes $(D_{initial} + 90°) - 180° = D_{initial} - 90°$.
- Turn 3: She turns 135° anticlockwise. Her direction becomes $(D_{initial} - 90°) - 135° = D_{initial} - 225°$.
- Turn 4: She turns 180° clockwise. Her final direction is $(D_{initial} - 225°) + 180° = D_{initial} - 45°$.
Calculating the Net Turn Effect
We can sum up all the clockwise and anticlockwise turns to find the net change in direction:
- Total Clockwise Rotation = 90° + 180° = 270°
- Total Anticlockwise Rotation = 180° + 135° = 315°
- Net Turn = Total Clockwise Turns - Total Anticlockwise Turns
- Net Turn = 270° - 315° = -45°
A net turn of -45° is equivalent to a 45° turn anticlockwise.
So, Zainab's final direction ($D_{final}$) is related to her initial direction ($D_{initial}$) by:
$$ D_{final} = D_{initial} + (\text{Net Turn}) $$
$$ D_{final} = D_{initial} - 45° $$
Finding the Initial Direction
We know that Zainab's final direction is North-East.
Using our angle convention, North-East corresponds to 45°.
So, we have:
$$ 45° = D_{initial} - 45° $$
To find $D_{initial}$, we rearrange the equation:
$$ D_{initial} = 45° + 45° $$
$$ D_{initial} = 90° $$
The direction that corresponds to an angle of 90° (when North is 0°) is East.
Directional Summary
Zainab's sequence of turns resulted in a net movement of 45° anticlockwise from her starting position. Since she ended up facing North-East (45°), her initial direction must have been 45° clockwise from North-East, which is East (90°).
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Important Questions from Direction and Distance
What is the direction of U with respect to P?
What is the shortest distance between R and T?
If a person walks 3m towards the North from point V, takes a right turn and walks for another 15m, which of the following points would he reach?
In which direction is Amit facing at point F?
What is the distance between the starting point and the end point?
