SSCE HSC Mathematics Extension 1 Scalar product of vectors: For the two vectors \( \overlin

For the two vectors $ \overline{OA} $ and $ \overline{OB} $ it is known that $ \overline{OA} \cdot \overline{OB} < 0 $ - HSC - SSCE Mathematics Extension 1 - Question 5 - 2021 - Paper 1

Question 5

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For the two vectors $ \overline{OA} $ and $ \overline{OB} $ it is known that $ \overline{OA} \cdot \overline{OB} < 0 $. Which of the following statements MUS... show full transcript

Worked Solution & Example Answer:For the two vectors $ \overline{OA} $ and $ \overline{OB} $ it is known that $ \overline{OA} \cdot \overline{OB} < 0 $ - HSC - SSCE Mathematics Extension 1 - Question 5 - 2021 - Paper 1

Step 1

A. Either, $ \overline{OA} $ is negative and $ \overline{OB} $ is positive, or, $ \overline{OA} $ is positive and $ \overline{OB} $ is negative.

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Answer

This statement cannot be concluded from the condition ( \overline{OA} \cdot \overline{OB} < 0 ). The dot product being negative indicates that the vectors point in opposite directions but does not strictly determine their individual positivity or negativity.

Step 2

B. The angle between $ \overline{OA} $ and $ \overline{OB} $ is obtuse.

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Answer

This statement must be true. The condition ( \overline{OA} \cdot \overline{OB} < 0 ) indicates that the angle ( \theta ) between the two vectors is greater than 90 degrees, making it obtuse, since ( \overline{OA} \cdot \overline{OB} = |\overline{OA}| |\overline{OB}| imes ext{cos} \theta).

Step 3

C. The product $ |\overline{OA}| |\overline{OB}| $ is negative.

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Answer

This statement is false. The product of the magnitudes of two vectors ( |\overline{OA}| ) and ( |\overline{OB}| ) is always non-negative, as both magnitudes are positive.

Step 4

D. The points O, A and B are collinear.

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Answer

This statement is not necessarily true. The condition given does not imply collinearity. The vectors can be positioned such that they form an obtuse angle without being collinear.

For the two vectors \( \overline{OA} \) and \( \overline{OB} \) it is known that \( \overline{OA} \cdot \overline{OB} < 0 \) - HSC - SSCE Mathematics Extension 1 - Question 5 - 2021 - Paper 1

Worked Solution & Example Answer:For the two vectors \( \overline{OA} \) and \( \overline{OB} \) it is known that \( \overline{OA} \cdot \overline{OB} < 0 \) - HSC - SSCE Mathematics Extension 1 - Question 5 - 2021 - Paper 1

A. Either, \( \overline{OA} \) is negative and \( \overline{OB} \) is positive, or, \( \overline{OA} \) is positive and \( \overline{OB} \) is negative.

B. The angle between \( \overline{OA} \) and \( \overline{OB} \) is obtuse.

C. The product \( |\overline{OA}| |\overline{OB}| \) is negative.

D. The points O, A and B are collinear.

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