Photo AI
Consider any three-dimensional vectors $ extbf{a} = extbf{O} extbf{A}$, $ extbf{b} = extbf{O} extbf{B}$ and $ extbf{c} = extbf{O} extbf{C}$ that satisfy these three conditions $$ extbf{a} ullet extbf{b} = 1$$ $$ extbf{b} ullet extbf{c} = 2$$ $$ extbf{c} ullet extbf{a} = 3$$ Which of the following statements about the vectors is true? A - HSC - SSCE Mathematics Extension 2 - Question 10 - 2023 - Paper 1
Question 10
Consider any three-dimensional vectors $ extbf{a} = extbf{O} extbf{A}$, $ extbf{b} = extbf{O} extbf{B}$ and $ extbf{c} = extbf{O} extbf{C}$ that satisfy these thr... show full transcript
Worked Solution & Example Answer:Consider any three-dimensional vectors $ extbf{a} = extbf{O} extbf{A}$, $ extbf{b} = extbf{O} extbf{B}$ and $ extbf{c} = extbf{O} extbf{C}$ that satisfy these three conditions $$ extbf{a} ullet extbf{b} = 1$$ $$ extbf{b} ullet extbf{c} = 2$$ $$ extbf{c} ullet extbf{a} = 3$$ Which of the following statements about the vectors is true? A - HSC - SSCE Mathematics Extension 2 - Question 10 - 2023 - Paper 1
Step 1
A. Two of a, b and c could be unit vectors.
Answer
To determine whether two of the vectors could be unit vectors, we need to analyze the given conditions. The inner product extbf{a} ullet extbf{b} = 1 suggests that the angle between them is such that their magnitudes and the cosine of the angle yield 1. Since unit vectors have a magnitude of 1, and the dot product of two unit vectors cannot exceed 1, statement A is plausible; however, it does not necessarily hold true for all configurations.
Step 2
B. The points A, B and C could lie on a sphere centred at O.
Answer
The condition that points A, B, and C lie on a sphere centered at O can be confirmed through the use of the inner products. If the three vectors adhere to the relationship of a sphere, then there exists that radius such that these points are equidistant from O. Given that the vectors have fixed inner products, it is indeed possible for these points to lie on a sphere centered at O. This statement is true.
Step 3
C. For any three-dimensional vector a, vectors b and c can be found so that a, b and c satisfy these three conditions.
Answer
This statement suggests that for any arbitrary vector , there exist vectors and fulfilling the prescribed dot product conditions. However, given those constraints, it may not always be feasible to find such vectors for every choice of . Thus, this statement is not universally true.
Step 4
D. ∀ a, b and c satisfying the conditions, ∃ r, s and t such that r, s and t are positive real numbers and r a + s b + t c = 0.
Answer
This statement implies a specific linear combination of the vectors leading to a zero vector, considering positive scalar multipliers. However, if all vectors are non-zero and maintain their inner product relationships, it is unlikely that a linear combination non-trivially sums to zero under positive constraints. Hence, this statement is false.