James travels 200m north then 350 south west. What is the bearing to the start point from the finish point?

Make a sketch.

Use the cosine law to find the distance between start and finish points.
Then use the sine law to find the missing angles.
Simple geometry will then let you find the "bearing" angle.

Let me know your answers to the intermediate steps.

To find the bearing from the finish point back to the start point, we can use the concept of vectors. A vector represents both the magnitude and direction of a displacement.

In this case, James first travels 200m north, which we can represent as a vector A. Then, he travels 350m southwest, which we can represent as a vector B.

To find the bearing from the finish point back to the start point, we need to find the resultant vector of A and B, which represents the net displacement from the start point to the finish point.

To visualize this, we can draw a diagram. Start by drawing a horizontal line to represent the x-axis and a vertical line to represent the y-axis. Mark the origin (0,0) as the starting point.

Now, draw vector A representing the 200m north. Since north is in the positive y-direction, vector A will point directly up from the origin.

Next, draw vector B representing the 350m southwest. Since southwest is a combination of south (negative y-direction) and west (negative x-direction), vector B will point downwards and to the left.

To calculate the resultant vector, we can add vectors A and B. To do this, we can use trigonometry and the Pythagorean theorem. Split vector B into its x and y components, using the southwest angle.

Now, calculate the x-component of B: Bx = 350m * cos(45°) = 350m / √2 ≈ -247.5m

And calculate the y-component of B: By = 350m * sin(45°) = 350m / √2 ≈ -247.5m

Add the x-components and y-components separately:

Ax + Bx = 0m + (-247.5m) = -247.5m
Ay + By = 200m + (-247.5m) = -47.5m

So, the resultant vector R has an x-component of -247.5m and a y-component of -47.5m.

To find the magnitude of the resultant vector, use the Pythagorean theorem:
|R| = √((-247.5m)^2 + (-47.5m)^2) ≈ 254m

Now, to find the bearing from the finish point back to the start point, we use the inverse tangent function. Take the arctan of (-47.5m / -247.5m) and convert the result to degrees.

Bearing = atan(-47.5m / -247.5m) ≈ 11.28°

Therefore, the bearing from the finish point back to the start point is approximately 11.28°.