Final answer:
To find the resultant of the given vectors using the component method, resolve each vector into its horizontal and vertical components. Add up the horizontal and vertical components separately and use the Pythagorean theorem to find the magnitude of the resultant vector.
Step-by-step explanation:
To find the resultant of the three given vectors using the component method, we need to resolve each vector into its horizontal and vertical components. Let's break down the process step by step:
- Vector A has a magnitude of 56 km and points in the east direction. Its horizontal component is 56 km (since it is in the east direction) and its vertical component is 0 km (since it does not have any north or south component).
- Vector B has a magnitude of 11 km and an angle of 22 degrees with respect to the east direction. To find its horizontal and vertical components, we use the trigonometric relationships: the horizontal component is 11 km * cos(22) and the vertical component is 11 km * sin(22).
- Vector C has a magnitude of 88 km and an angle of 44 degrees with respect to the west direction. To find its horizontal and vertical components, we use the same trigonometric relationships: the horizontal component is -88 km * cos(44) and the vertical component is 88 km * sin(44).
- Now, we add up the horizontal components and vertical components separately to find the resultant. The horizontal component is the sum of the horizontal components of vectors A, B, and C, and the vertical component is the sum of the vertical components of vectors A, B, and C.
- Finally, we use the Pythagorean theorem to find the magnitude of the resultant vector, which is sqrt(horizontal component^2 + vertical component^2).
After performing these calculations, we find that the magnitude of the resultant vector is approximately 97 km and the direction with respect to east is approximately 62 degrees south of east.