Calculating 3-D Vectors: Mountain Summit from Base Camp

[physicsss]

The summit of a mountain, 2000 m above base camp, is measured on a map to be 4000 m horizontally from the camp in a direction 40.0° west of north. Choose the x-axis east, y-axis north, and z axis up. What are the x, y, and z components of the displacement vector from camp to summit?

How do I treat the z vector...what does a picture of the vectors look like?

 

[Janitor]

The way I see it, there is just one vector (the "displacement vector") that you have to worry about. I wonder if the z component of this vector is what you mean when you say "the z vector." The tail of the displacement vector lies at the camp, and the tip of the vector lies at the summit.

The z component of this vector is actually the easiest to write down. In fact, no calculation is needed to figure the z component out, because it is given in the problem statement.

Additional hints: the x component will be negative, and the y component will be positive. These two components can be figured out from some trigonometry involving a right triangle with an angle of 40 degrees and a hypotenuse with a length of 4000. In particular, see what you can do with sine and cosine.

 

[physicsss]

Would cos 40*4000 give me the x-component and sin 40 *4000 the y component? And also, how do I find out the length of the displacement?

 

[HallsofIvy]

Would cos 40*4000 give me the x-component and sin 40 *4000 the y component? And also, how do I find out the length of the displacement?

Be careful about the directions: you are told that the positive x-axis is east and that the direction is 40 degrees west of north. Draw a picture and look at it carefully!

Of course, the z component is just that 2000 m height. You have a right triangle with legs of length 2000 and 4000. What is the length of the hypotenuse?

 

[Janitor]

To echo HallsofIvy, be careful about which component you calculate with cosine and which with sine. And don't forget the negative sign where appropriate. As Halls says, you can look at the displacement length as the hypotenuse of a right triangle in a vertical plane, where you know the base and the side. Or alternatively, you can use the fact that you will have figured out all three Cartesian components of the vector, and then square each of the three, add these three (positive!) squares together, and take the square root.

 

[physicsss]

So since it is 40 degrees west of North, I should subtract it from 90 first?
So I would get -cos 60*4000 as the x-component and sin 60*4000 as the y component?

 

[HallsofIvy]

Yes, that's correct.

 

[Janitor]

Be aware that the complement of a 40 degree angle is a 50 degree angle, not a 60 degree angle.