The angle between two unit vectors $\mathbf{a}$ and $\mathbf{b}$ is $\theta$ and $|\mathbf{a} + \mathbf{b}| < 1$ - HSC - SSCE Mathematics Extension 1 - Question 8 - 2022 - Paper 1
Question 8
The angle between two unit vectors $\mathbf{a}$ and $\mathbf{b}$ is $\theta$ and $|\mathbf{a} + \mathbf{b}| < 1$.
Which of the following best describes the possible... show full transcript
Worked Solution & Example Answer:The angle between two unit vectors $\mathbf{a}$ and $\mathbf{b}$ is $\theta$ and $|\mathbf{a} + \mathbf{b}| < 1$ - HSC - SSCE Mathematics Extension 1 - Question 8 - 2022 - Paper 1
Step 1
Determine the Condition of the Vectors
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Answer
The condition ∣a+b∣<1 implies that the vectors must form an angle of more than 0 degrees (or 0 radians) but less than 180 degrees (or π radians). This is because if the angle were 0, the vectors would point in the same direction, and their sum would be ∣a∣+∣b∣=2, violating the condition. Conversely, if the angle were 180 degrees, their sum would be ∣a∣+∣b∣=0, which is also not allowed.
Step 2
Analyze the Angles and Select the Correct Range
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Answer
Considering the cosine of the angle between the two vectors, we know that:
cos(θ)=a⋅b≤1.
Since ∣a+b∣<1, we establish that:
θ>32π to ensure the angle is obtuse.
Thus, the only range that satisfies all conditions is: 32π<θ≤π.
