A methane molecule can be represented by four equally spaced hydrogen atoms and

Question

A methane molecule can be represented by four equally spaced hydrogen atoms and one carbon atom at the center. Let A = (0,0,0), B = (2,0,0), C= (1, sqrt(3), 0), and D = (1,1/sqrt(3), 2sqrt(2/3)) be the locations of the hydrogen atoms.

(a) Show A,B,C, and D are all the same distance from each other.

(b) Find the position of the central carbon atom P = (x,y,z) which is equidistant from A,B,C, and D by setting up and solving three equations in x,y, and z.

(c) Use the dot product to find the angle APB.

Explanation / Answer

a)To show that they are same distance from each other

lets show AB= AC=BC

AB= 2

AC= 2

BC= 2

By using distance formula

b) Position of carbon is (x,y,z)

x^2+y^2+z^2 = x-2^2+y^2+z^2

x^2 = x-2^2

so x = 1

similarly we can find y and z

y= 1/2 sqrt3

z= - sqrt 3/4sqrt2

c)the angle of APB will be 109 degrees

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