Saturday, June 02, 2007

Two Multiple Choice Questions on Vectors

A methane molecule CH4 may be fitted in a cube of side 2a such that the C atom lies at the body centre and four H-atoms at non-adjacent corners of the cube as shown in figure. The angle between any two C-H bonds is
(a) 120º
(b) cos–1(1/3)
(c) 150º
(d) sin–1(1/3)
(e) cos–1(1/3)
This MCQ appeared in Kerala Engineering Entrance 2006 question paper. You are required to find the angle between the lines CH1 and CH2. Treat these lines as vectors. The coordinate axes are marked (by the question setter) in the figure with H3 as the origin.
The vectors CH1 and CH2 have X, Y and Z components of the same magnitude ‘a’ since the carbon atom C is at the centre of the cube of side 2a. These vectors can be written as
CH1 = a i +a j +a k
CH2 = +a i a j +a k
[Note that bold face letters represent vectors and i, j, k are unit vectors in the X, Y and Z directions respectively].
The magnitudes CH1 and CH2 of these vectors are the same, equal to a√3.
The angle between the vectors CH1 and CH2 is given by
Cos(CH1, CH2) = (CH1 . CH2)/ (CH1)(CH2) = (a2 a2 +a2)/ (a√3)( a√3)
= –(1/3)
The angle between the vectors, which is the bond angle required, is therefore cos–1(1/3).
Now, consider the following question. This is a simple one, but similar questions are often seen in entrance test papers.
Angle (in rad) made by the vector j + √3 k with the Y-axis is
(a) π/8 
(b) π/6 
(c) π/4 
(d) π/3 
(e) π/2
You should remember that the direction cosines (the cosines of the angles between coordinate axes and   the vector itself) are given by Cos(A,x) = Ax/A, Cos(A,y) = Ay/A and Cos(A,z) = Az/A.
Since the Y-component has magnitude 1 and the vector has magnitude 2, the cosine of the angle between the Y-axis and the given vector is ½. The angle therefore is π/3.

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