A spider crawls 1 m due north , 2 m due east and then climbs 3 m vertically on a wall . What is the magnitude of the resultant dispalcemen ?

Explanation

To find the magnitude of the resultant displacement of the spider, we can break down the problem step by step.

### Step 1: Identify the movements of the spider
The spider makes three movements:
1. Crawls 1 meter due north.
2. Crawls 2 meters due east.
3. Climbs 3 meters vertically.

### Step 2: Represent the movements in a coordinate system
We can represent the spider's movements in a 3D coordinate system:
- North direction can be represented as the positive Y-axis.
- East direction can be represented as the positive X-axis.
- Vertical direction can be represented as the positive Z-axis.

Thus, we can assign:
- Displacement in the Y direction (north) = 1 m
- Displacement in the X direction (east) = 2 m
- Displacement in the Z direction (vertical) = 3 m

### Step 3: Use the Pythagorean theorem for 3D displacement
To find the resultant displacement \( S \), we use the formula for the magnitude of the resultant vector in three dimensions:
\[ S = \sqrt{S_x^2 + S_y^2 + S_z^2} \]
where:
- \( S_x \) = displacement in the X direction = 2 m
- \( S_y \) = displacement in the Y direction = 1 m
- \( S_z \) = displacement in the Z direction = 3 m

### Step 4: Substitute the values into the formula
Substituting the values into the formula:
\[ S = \sqrt{(2)^2 + (1)^2 + (3)^2} \]
Calculating each term:
\[ S = \sqrt{4 + 1 + 9} \]

### Step 5: Calculate the sum inside the square root
\[ S = \sqrt{14} \]

### Step 6: Find the numerical value of the resultant displacement
Calculating the square root:
\[ S \approx 3.74 \text{ m} \]

### Conclusion
The magnitude of the resultant displacement of the spider is approximately **3.74 meters**.

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