Question

If $a$, $b$, $c$ are the position vectors of $A$, $B$, $C$ respectively such that $3\mathbf{a} + 4\mathbf{b} - 7\mathbf{c} = \mathbf{0}$, then $C$ divides $AB$ in the ratio

• A. 4:3
• B. 4:7
• C. 3:7
• D. 3:4

Hint:

Section formula: $\mathbf{c} = \dfrac{n\mathbf{a}+m\mathbf{b}}{m+n}$ means $C$ divides $AB$ internally in ratio $m:n$ (coefficient of $\mathbf{b}$ : coefficient of $\mathbf{a}$).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Use the section formula: if $C$ divides $AB$ in ratio $m:n$, then $\mathbf{c} = \dfrac{m\mathbf{b}+n\mathbf{a}}{m+n}$.
Step 2: Detailed Explanation:
$3\mathbf{a}+4\mathbf{b} = 7\mathbf{c} \Rightarrow \mathbf{c} = \dfrac{3\mathbf{a}+4\mathbf{b}}{7}$.
Comparing with $\mathbf{c}=\dfrac{n\mathbf{a}+m\mathbf{b}}{m+n}$: $n=3$, $m=4$, so $C$ divides $AB$ in ratio $m:n=4:3$.
Step 3: Final Answer:
$C$ divides $AB$ in the ratio $4:3$.