A person leaves home and walks 3 miles west, then 4 miles southwest. How far from home is she? miles In what direction must she walk to head directly home? degrees North of East Calculator Check Answer

Step 1 ：The person's movement can be represented as vectors. The first movement is 3 miles west, which we can represent as a vector (-3, 0). The second movement is 4 miles southwest. Southwest can be represented as a vector with equal parts south and west, so we can represent this as a vector (-2, -2).

Step 2 ：To find the total displacement, we can add these vectors together to get a displacement vector of (-5, -2).

Step 3 ：To find the distance from home, we can use the Pythagorean theorem, which states that the square of the hypotenuse (the distance from home) is equal to the sum of the squares of the other two sides (the x and y components of the displacement). Using this theorem, we find that the distance from home is approximately \( \sqrt{(-5)^2 + (-2)^2} = \boxed{5.39} \) miles.

Step 4 ：To find the direction she must walk to head directly home, we can use the arctangent function to find the angle between the displacement vector and the positive x-axis. This will give us the direction in degrees counterclockwise from east. To convert this to degrees north of east, we can subtract the result from 90. Using this method, we find that she must walk approximately \( 90 - \arctan\left(\frac{-2}{-5}\right) = \boxed{68.20} \) degrees North of East to head directly home.