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) = A_x/A, Cos(A,y) = A_y/A and Cos(A,z) = A_z/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.