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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
Question 10
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. What is the value... show full transcript
Worked Solution & Example Answer: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
Step 1
Identify the conditions on the vectors
Answer
Given the conditions that the vectors a, b, and c are unit vectors ( |a| = |b| = |c| = 1) and are mutually orthogonal (a ⊥ b, b ⊥ c), it follows that the dot products are:
[ a \cdot b = 0 ]
[ b \cdot c = 0 ]
This implies that the angle between a and b is 90 degrees, therefore, their cosine is 0.
Step 2
Determine the angle θ and its cosine
Answer
To find the angle θ between vector a and the resultant vector b + c, we can calculate:
[ \cos \theta = \frac{a \cdot (b + c)}{|a| |b + c|} ]
Since |a| = 1 and |b + c| can be computed as:
[ |b + c| = \sqrt{|b|^2 + |c|^2} = \sqrt{1^2 + 1^2} = \sqrt{2} ]
This leads us to:
[ \cos \theta = \frac{0}{\sqrt{2}} = 0 ]