Q.69. Vectors

Question 69.

If the unit vectors e1 and e2 are inclined at an angle 2θ and

l e1 vector - e2 vector l  < 1, then prove that for θ ∈ [0, π],

θ may lie in the interval [(5π)/6, π].

Answer 69.
Let e1 vector be along x-axis
and e2 vector be at an angle 2θ in the clockwise direction from e1 vector.
As both are unit vectors,
e1 vector = (1, 0) and e2 vector = (cos2θ, sin2θ)
Now, l e1 vector - e2 vector l < 1
=> lcos2θ - 1, sin2θl < 1
=> (cos2θ - 1)^2 + sin^2 2θ < 1
=> cos^2 2θ - 2cos2θ + 1 + sin^2 θ < 1
=> 1 - 2cos2θ < 0
=> cos2θ > 1/2
=> 0 < 2θ < π/3 or 5π/3 < 2θ < 2π
From the second condition,
5π/6 < θ < π.

Further explanation:
0 < θ < π => 0 < 2θ < 2π
cosine function is positive in the first and the fourth quadrant
Hence, cos2θ > 1/2
=> (1/2) < cos2θ < 1
=> cos(π/3) < cos2θ < cos0
Now, as cosine is a decreasing function in the first quadrant
=> π/3 > 2θ > 0
=> 0 < 2θ < π/3
=> 0 < θ < π/6

Similarly, for the fourth quadrant,
cos2θ > 1/2
=> (1/2) < cos2θ < 1
=> cos(5π/3) < cos2θ < cos2π
Now, as cosine is an increasing function in the fourth quadrant
=> 5π/3 < 2θ < 2π
=> 5π/6 < θ < π.

LINK to YA!