Where to camp in Rocky Mountain?

The summit of a mountain, 2068 m above base camp, is measured on a map to be 4571 m horizontally from the camp?

• The summit of a mountain, 2068 m above base camp, is measured on a map to be 4571 m horizontally from the camp in a direction 32.4° west of north. Choose the x axis east, y axis north, and z axis up. What are the components of the displacement vector from camp to summit? X__________m Y__________m Z__________m What is the magnitude?

• Answer:

  I like how the problem says "choose" your axis then tells you what to use. So we have a three dimensional vector, which is superb because we live in the third dimension so we're right at home. If this were a four-dimensional problem we might have been in trouble! OK, so we are asked to find the magnitudes of the components of this 3-d vector. We're given pieces of information, and are expected to use those pieces to fill in the rest of the blanks. First let's visualize the problem. We'll pretend we're a bird, in this case an unladen African Swallow so we can get up high enough. We're looking down on the mountain and camp site. We are oriented to the North because it's summer, and that's where birds fly for the summer in search of coconuts. So we've got our x-axis as E-W, our y-axis as N-S, and our z-axis as up and down (elevation). The problem gives us a few of the blanks already. The z-component is 2068m. The x-component is... the problem worded that kind of tricky. I'm not entirely sure if they're giving us the actual x-component or the hypotenuse (straight line path). I think since they gave us an angle, they're talking about the hypotenuse and expect us to derive the x and y components from that angle and magnitude. Sounds like something a devilish teacher would do, so let's do it. It's best to draw a picture and know your trig functions to be sure you're picking the right ones. In this case, since the angle is measured from North towards West, the components will be: x = 4571 sin(32.4°) = 2449 y = 4571 cos(32.4°) = 3859 That kinda makes sense that y is bigger than x since the angle was closer to North (y) than West (x), so it's an indicator that we picked the right functions. OK, so now we know all the components, huzzah! x = 2449m y = 3859m z = 2068m Now, the magnitude of that vector is according to the Pythagorean Theorem, meaning it is the square root of the sum of the squares. Or some other such nonsense. v² = 2449² + 3859² + 2068² v² = 5,997,601 + 14,891,881 + 4,276,624 v² = 25,166,106 v = 5,017 There's a lot of big numbers there, so I might have made a typo somewhere. But the answer sounds right. The magnitude of a vector will always be bigger than it's biggest component, but usually in the same ballpark. Both of those things are true here, so it looks right.