Finding the Bird's Final Direction from its Nest

This problem involves tracking the movement of a bird from a starting point (its nest) through a series of displacements in different directions. We need to determine the final direction of the bird relative to its starting point.

Let's break down the bird's flight into individual steps:

1. First movement: 25 m towards the North-East.
2. Second movement: A right turn from the current direction and flew 7 m.
3. Third movement: Another right turn from the current direction and flew 75 m.

We can visualize these movements or use a coordinate system approach. Let the nest be at the origin (0,0). We'll consider North as the positive y-axis, East as the positive x-axis, South as the negative y-axis, and West as the negative x-axis.

Analyzing Each Movement and Turns

Understanding the direction of turns is crucial:

• North-East direction is typically considered to be 45 degrees from the East towards the North.
• A right turn means turning 90 degrees clockwise from the current direction.

Let's determine the direction for each leg of the journey:

• Movement 1: 25 m North-East. The direction is 45 degrees counter-clockwise from the positive x-axis (East).
• Movement 2: Right turn from North-East. Turning 90 degrees clockwise from North-East (45 degrees) results in a direction of $45^\circ - 90^\circ = -45^\circ$ from the positive x-axis. This direction is South-East. The bird flew 7 m in the South-East direction.
• Movement 3: Right turn from South-East. Turning 90 degrees clockwise from South-East (-45 degrees) results in a direction of $-45^\circ - 90^\circ = -135^\circ$ from the positive x-axis. This direction is South-West. The bird flew 75 m in the South-West direction.

Using Vector Analysis to Find Final Position

We can represent each movement as a vector. The displacement from the nest is the sum of these vectors.

We'll use trigonometry to find the x and y components of each movement vector. Remember $\cos(45^\circ) = \sin(45^\circ) = \frac{1}{\sqrt{2}}$.

• Movement 1 (25 m North-East): $\vec{v_1} = (25 \cos 45^\circ, 25 \sin 45^\circ) = \left(\frac{25}{\sqrt{2}}, \frac{25}{\sqrt{2}}\right)$
• Movement 2 (7 m South-East): $\vec{v_2} = (7 \cos (-45^\circ), 7 \sin (-45^\circ)) = (7 \cos 45^\circ, -7 \sin 45^\circ) = \left(\frac{7}{\sqrt{2}}, -\frac{7}{\sqrt{2}}\right)$
• Movement 3 (75 m South-West): South-West corresponds to an angle of 225 degrees (or -135 degrees) from the positive x-axis. $\vec{v_3} = (75 \cos 225^\circ, 75 \sin 225^\circ) = (75 (-\cos 45^\circ), 75 (-\sin 45^\circ)) = \left(-\frac{75}{\sqrt{2}}, -\frac{75}{\sqrt{2}}\right)$

The total displacement vector $\vec{D}$ from the nest is the sum of these vectors:

$\vec{D} = \vec{v_1} + \vec{v_2} + \vec{v_3}$

Let's calculate the x and y components of the total displacement:

X-component: $D_x = \frac{25}{\sqrt{2}} + \frac{7}{\sqrt{2}} - \frac{75}{\sqrt{2}} = \frac{25 + 7 - 75}{\sqrt{2}} = \frac{32 - 75}{\sqrt{2}} = \frac{-43}{\sqrt{2}}$

Y-component: $D_y = \frac{25}{\sqrt{2}} - \frac{7}{\sqrt{2}} - \frac{75}{\sqrt{2}} = \frac{25 - 7 - 75}{\sqrt{2}} = \frac{18 - 75}{\sqrt{2}} = \frac{-57}{\sqrt{2}}$

The final position of the bird relative to the nest is at coordinates $\left(-\frac{43}{\sqrt{2}}, -\frac{57}{\sqrt{2}}\right)$.

Determining the Final Direction

To find the direction of the bird from its nest (the origin), we look at the signs of the x and y components of the final displacement vector:

• The x-component ($D_x = -\frac{43}{\sqrt{2}}$) is negative. This indicates the bird is to the West of the nest.
• The y-component ($D_y = -\frac{57}{\sqrt{2}}$) is negative. This indicates the bird is to the South of the nest.

When both the x and y coordinates relative to the origin are negative, the position lies in the South-West quadrant.

Therefore, the bird is now in the South-West direction from its nest.

This confirms the direction obtained by tracking the steps.

Summary of Movements and Directions

After all the movements, the bird's final position relative to the nest has negative x and negative y components, placing it in the South-West direction.

Revision Table: Understanding Directional Changes

Additional Information: Directional Reasoning and Vectors

Directional reasoning questions often appear in aptitude tests. They test your ability to follow a path based on directions and turns.

• Cardinal Directions: North, South, East, West.
• Intercardinal Directions: North-East, South-East, South-West, North-West.
• Each main direction is 90 degrees apart. Each intercardinal direction is 45 degrees from the nearest cardinal directions.
• A 'right turn' is always a 90-degree clockwise rotation. A 'left turn' is a 90-degree counter-clockwise rotation.
• For complex paths, breaking down movements into vectors (using x and y components) can help find the net displacement and final direction from the start point, especially when distances are involved and a simple diagram might not be precise enough.

In this problem, although the distances varied, the final direction from the nest was determined by the cumulative effect of the large South-West movement dominating the earlier North-East and South-East movements in terms of displacement components.