a man walks 5 miles due east and 7 miles southeast. how far, and in what direction is he from his starting point?
To find the distance and direction from the starting point, we can use vector addition. Let's break it down step by step:
1. Draw a diagram: Start by drawing a line to represent the man's initial 5 miles due east. Then, draw a second line at a 45-degree angle to represent the 7 miles southeast.
2. Analyze the diagram: The first line is directly east, so we can call it "5i" (where "i" represents the unit vector in the east direction). The second line is at a 45-degree angle southeast, so we can call it "7(cos45°i - sin45°j)" (where "j" represents the unit vector in the south direction).
3. Calculate the vector sum: To find the total displacement, add the two vectors together:
Total displacement = 5i + 7(cos45°i - sin45°j)
4. Simplify the expression: Combining like terms, we get:
Total displacement = (5 + 7cos45°)i - 7sin45°j
5. Calculate the magnitude and direction: The magnitude of the total displacement is given by the formula:
Magnitude = √(x^2 + y^2), where x and y are the coefficients of i and j, respectively.
Magnitude = √((5 + 7cos45°)^2 + (-7sin45°)^2)
The direction of the displacement can be found using the formula:
Direction = arctan(y / x), where x and y are the coefficients of i and j, respectively.
Direction = arctan((-7sin45°) / (5 + 7cos45°))
6. Calculate the values: Plugging in the values, we get:
Magnitude ≈ 9.8 miles
Direction ≈ 174 degrees east of north
Therefore, the man is approximately 9.8 miles away from his starting point in a direction that is approximately 174 degrees east of north.