Question
Download Solution PDFThe position vectors of three points A, B and C are a, b and c respectively such that \(\vec{3a}-\vec{4b}+\vec{c}=\vec{0}\) What is AB:BC equal to?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFCalculation:
Given,
\( 3\vec{a} - 4\vec{b} + \vec{c} = 0 \)
\( \vec{c} = 4\vec{b} - 3\vec{a} \)
The vector\(\overrightarrow{AB} \) is:
\( \overrightarrow{AB} = \vec{b} - \vec{a} \)
The vector \(\overrightarrow{BC} \) is:
\( \overrightarrow{BC} = \vec{c} - \vec{b} \)
Substituting \(\vec{c} = 4\vec{b} - 3\vec{a} \):
\( \overrightarrow{BC} = (4\vec{b} - 3\vec{a}) - \vec{b} \)
\( \overrightarrow{BC} = 3\vec{b} - 3\vec{a} \)
Step 4: Now, \(\overrightarrow{BC} = 3(\vec{b} - \vec{a}) \), which gives:
\( AB : BC = 1 : 3 \)
∴ The correct ratio is AB : BC = 1 : 3 ,
Hence, the correct answer is Option 2.
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