Consider any three-dimensional vectors $ \mathbf{a} = \overrightarrow{OA}, \mathbf{b} = \overrightarrow{OB} \text{ and } \mathbf{c} = \overrightarrow{OC} $ that satisfy these three conditions $ \mathbf{a} \cdot \mathbf{b} = 1 \\ \mathbf{b} \cdot \mathbf{c} = 2 \\ \mathbf{c} \cdot \mathbf{a} = 3 $ Which of the following statements about the vectors is true? A. Two of $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ could be unit vectors. B. The points $A, B$ and $C$ could lie on a sphere centred at $O$. C. For any three-dimensional vector $\mathbf{a}$, vectors $\mathbf{b}$ and $\mathbf{c}$ can be found so that $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ satisfy these three conditions. D. $\forall \mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ satisfying the conditions, $\exists r, s, t$ such that $r, s$ and $t$ are positive real numbers and $r\mathbf{a} + s\mathbf{b} + t\mathbf{c} = 0$.

SSCE HSC Mathematics Extension 2 Angle between vectors: Consider any three-dimensional v

Consider any three-dimensional vectors $ \mathbf{a} = \overrightarrow{OA}, \mathbf{b} = \overrightarrow{OB} \text{ and } \mathbf{c} = \overrightarrow{OC} $ that satisfy these three conditions $ \mathbf{a} \cdot \mathbf{b} = 1 \\ \mathbf{b} \cdot \mathbf{c} = 2 \\ \mathbf{c} \cdot \mathbf{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

$Consider-any-three-dimensional-vectors---$-\mathbf{a}-=-\overrightarrow{OA},-\mathbf{b}-=-\overrightarrow{OB}-\text{-and-}-\mathbf{c}-=-\overrightarrow{OC}-$--that-satisfy-these-three-conditions--$-\mathbf{a}-\cdot-\mathbf{b}-=-1-\\--\mathbf{b}-\cdot-\mathbf{c}-=-2-\\--\mathbf{c}-\cdot-\mathbf{a}-=-3-$--Which-of-the-following-statements-about-the-vectors-is-true?--A-HSC-SSCE Mathematics Extension 2-Question 10-2023-Paper 1.png$

Consider any three-dimensional vectors $ \mathbf{a} = \overrightarrow{OA}, \mathbf{b} = \overrightarrow{OB} \text{ and } \mathbf{c} = \overrightarrow{OC} $ that s... show full transcript

Worked Solution & Example Answer:Consider any three-dimensional vectors $ \mathbf{a} = \overrightarrow{OA}, \mathbf{b} = \overrightarrow{OB} \text{ and } \mathbf{c} = \overrightarrow{OC} $ that satisfy these three conditions $ \mathbf{a} \cdot \mathbf{b} = 1 \\ \mathbf{b} \cdot \mathbf{c} = 2 \\ \mathbf{c} \cdot \mathbf{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 $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ could be unit vectors.

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The dot product condition $\mathbf{a} \cdot \mathbf{b} = 1$ suggests that both vectors could be unit vectors. If both $\mathbf{a}$ and $\mathbf{b}$ are unit vectors, their dot product would indeed be 1, indicating this statement could be true. However, this does not hold for $\mathbf{c}$ with the given conditions.

Step 2

B. The points $A, B$ and $C$ could lie on a sphere centred at $O$.

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This statement is plausible as points $A$ , $B$ , and $C$ can be positioned such that $\overrightarrow{OA}, \overrightarrow{OB}, \overrightarrow{OC}$ satisfy the conditions of angles and lengths, suggesting they can be on a sphere with center $O$ . Thus, this statement is true.

Step 3

C. For any three-dimensional vector $\mathbf{a}$, vectors $\mathbf{b}$ and $\mathbf{c}$ can be found so that $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ satisfy these three conditions.

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This claim is more complex. The conditions establish specific relationships that might not hold true for arbitrary vectors $\mathbf{b}$ and $\mathbf{c}$ . Hence, without further constraints on the vectors, this statement is false.

Step 4

D. $\forall \mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ satisfying the conditions, $\exists r, s, t$ such that $r, s$ and $t$ are positive real numbers and $r\mathbf{a} + s\mathbf{b} + t\mathbf{c} = 0$.

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The statement suggests a linear relation between the vectors. Given the conditions, we cannot guarantee positive values for $r, s$ , and $t$ across all vectors. Therefore, this statement is false.

A. Two of $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ could be unit vectors.

B. The points $A, B$ and $C$ could lie on a sphere centred at $O$.

C. For any three-dimensional vector $\mathbf{a}$, vectors $\mathbf{b}$ and $\mathbf{c}$ can be found so that $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ satisfy these three conditions.

D. $\forall \mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ satisfying the conditions, $\exists r, s, t$ such that $r, s$ and $t$ are positive real numbers and $r\mathbf{a} + s\mathbf{b} + t\mathbf{c} = 0$.

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