From a starting point you drive 9 km in a straight line in a direction 1.5 radians east of north to reach the final point. How far in kilometers would you have to drive from the same starting point to arrive at the same final point if you first drove a distance directly north and then drove a distance directly east?

This is a problem of decomposing a vector into its components along the north and east directions. 

Given:
- Total distance traveled in a straight line: 9 km
- Direction: 1.5 radians east of north.

We can break the 9 km displacement into two components:
- A northward component (along the north direction)
- An eastward component (along the east direction)

The components can be calculated using trigonometric functions:

- The northward component is given by:  
  \( \text{Northward distance} = 9 \cdot \cos(1.5) \)
  
- The eastward component is given by:  
  \( \text{Eastward distance} = 9 \cdot \sin(1.5) \)

### Calculation:

1. \( \cos(1.5) \approx 0.0707 \)
2. \( \sin(1.5) \approx 0.9975 \)

Now, multiplying these by the total distance (9 km):

- Northward distance = \( 9 \cdot 0.0707 \approx 0.636 \, \text{km} \)
- Eastward distance = \( 9 \cdot 0.9975 \approx 8.977 \, \text{km} \)

So, you would have to drive:
- \( 0.636 \, \text{km} \) north,
- \( 8.977 \, \text{km} \) east,

to reach the same final point.

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Would you like further details on these calculations, or do you have any other questions?

Here are 5 questions that build on this problem:
1. What if the angle had been given in degrees instead of radians?
2. How would the distances change if the total distance was 15 km instead of 9 km?
3. How can you use the Pythagorean theorem to check the solution?
4. What would the distances be if the direction was due east instead of 1.5 radians east of north?
5. How would you find the direction if you were given the northward and eastward distances?

**Tip:** For angles in radians, \( \sin \) and \( \cos \) values are crucial for breaking vectors into components.

Understand how to decompose a vector of 9 km at an angle of 1.5 radians east of north into northward and eastward components. Learn the trigonometric approach using sine and cosine to find the distances. Suitable for students in Grades 10-12.

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