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The angle between two unit vectors \( a \) and \( b \) is \( \theta \) and \( |a + b| < 1 \) - HSC - SSCE Mathematics Extension 1 - Question 8 - 2022 - Paper 1
Question 8
The angle between two unit vectors \( a \) and \( b \) is \( \theta \) and \( |a + b| < 1 \). Which of the following best describes the possible range of values of \... show full transcript
Worked Solution & Example Answer:The angle between two unit vectors \( a \) and \( b \) is \( \theta \) and \( |a + b| < 1 \) - HSC - SSCE Mathematics Extension 1 - Question 8 - 2022 - Paper 1
Step 1
Determine the relationship between the vectors and their angle
Answer
Given that ( a ) and ( b ) are unit vectors, their magnitudes are ( |a| = |b| = 1 ). The condition ( |a + b| < 1 ) implies that the cosine of the angle between them must also be evaluated. The angle can be expressed using the dot product: ( a \cdot b = \cos(\theta) ).
Step 2
Analyze the implications of the given condition
Answer
From the condition ( |a + b| < 1 ), we can derive that the angle ( \theta ) must be greater than a certain range. Specifically, if both vectors point in the same direction, we would obtain the maximum resultant length of 2, so the angle must be constrained. This means ( \theta ) lies in the range that ensures the sum is less than 1.
Step 3
Select the correct answer based on the analysis
Answer
After analyzing the expressions derived from the conditions and the angle between the vectors, the correct statement regarding the possible range of values of ( \theta ) is that it fits into the range: ( \frac{2\pi}{3} < \theta < \pi ), which correlates to option D.