Finding height using the direction cosines

[janahan]

Sorry for the re-post

Homework Statement

http://books.google.ca/books?id=nYR...g direction cosines to measure height&f=false

The obove link has a coopy of the question as it is hard to describe.

If you find the link is too hard to read here is the data
Direction cosines:

of Rap(from point a to top of mountain):
cos theta x= .5179
cos theta y= .6906
cos theta z= .5048

of Rbp(from point b to top of mountain):
cos theta x=-.3743
cos theta y=.7486
cos theta z=.5472

b and a are 10000m apart. Find how high point p is.
It is on page 57 #2.82 (The mount everest question)

Homework Equations

The Attempt at a Solution

I attemted the question in a number of ways but can't seem to come to an answer. I tried Looking at it as two right angled triangles and using trig. I am not really sure what i can do with the direction cosines either.

 

[rl.bhat]

Draw a perpendicular PO on the base of the mountain. Let PD, PE and PF on x, y and z axis passing through A.
AD = AP*cos(α)
AE = AP*cos(β)
[cos(α) = 0.5179, cos(β) = 0.6906]
Now tan(OAD) = AE/AD. Find angle OAD. Similarly find angle OBD. from these two angles find the third angle AOB. Using sine rule in the triangle AOB, find Ao and BO.
cos(γ) = 0.5048 is given. From that find tan(γ) which is equal to PO/AO. AO is known. Find PO.

 

[tomcue]

I would approach this problem by first understanding the concept of direction cosines and how they relate to height measurement. Direction cosines are unit vectors that represent the direction of a line or vector in three-dimensional space. In this case, the direction cosines are being used to represent the direction from two points (a and b) to a third point (p) on a mountain.

To solve this problem, we can use the fact that the direction cosines are related to the angle between the line and each of the coordinate axes. We can also use the distance between points a and b (10,000m) to help us find the height of point p.

First, we can draw a diagram to visualize the problem. From point a, we can draw a line to the top of the mountain (point p) and label it as vector Rap. Similarly, from point b, we can draw a line to the top of the mountain and label it as vector Rbp.

Next, we can use the direction cosines to find the angles between these vectors and the coordinate axes. This will give us the angles between the lines Rap and Rbp and the horizontal plane. We can then use trigonometry to find the height of point p.

Using the given data, we can calculate the angles between the vectors and the horizontal plane:

θx = cos^-1(0.5179) ≈ 59.5°
θy = cos^-1(0.6906) ≈ 45.2°
θz = cos^-1(0.5048) ≈ 61.2°

Next, we can use the law of sines to find the angle between the lines Rap and Rbp:

sin(θx)/10,000 = sin(θy)/h

Where h is the height of point p. Solving for h, we get:

h = 10,000(sin(θy)/sin(θx)) ≈ 11,900m

Therefore, the height of point p is approximately 11,900m. We can also use the direction cosines of Rbp to double check our answer:

sin(θx)/10,000 = sin(θz)/h

Solving for h, we get the same answer of approximately 11,900m.

In conclusion, using direction cosines is a useful method for measuring height in three-dimensional space. By understanding the