Three unit vectors a, b and c in 3 dimensions, are to be chosen so that a ⊥ b, b ⊥ c and the angle θ between a and b + c is as small as possible - HSC - SSCE Mathematics Extension 2 - Question 10 - 2024 - Paper 1

To minimize the angle θ between vector a and the sum of vectors b and c, we can use the geometric property of unit vectors in three dimensions. Since a, b, and c are orthogonal, we find that:

1. The vector sum b + c is still a unit vector. We can express this sum as:

   $b + c = \sqrt{b^2 + c^2} = \sqrt{1 + 1} = \sqrt{2}$

2. The angle θ can then be calculated using the dot product. Since all vectors are unit vectors and orthogonal:

   cos(θ) = \frac{a ullet (b + c)}{|a| |b + c|} = \frac{0}{|b + c|} = 0

Given that the angle between a and (b + c) is minimized, the cosine of this angle is:

$cos(θ) = 0$

Therefore, the answer is:

Option A: 0.