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
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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 $\mathbf{a}$).
To determine the ratio in which point \( C \) divides the line segment \( AB \), given the condition \( 3\mathbf{a} + 4\mathbf{b} - 7\mathbf{c} = \mathbf{0} \), we can solve this using vector concepts.
Recall the section formula in the context of vectors. If a point \( C \) divides the line segment joining two points \( A \) and \( B \) with position vectors \( \mathbf{a} \) and \( \mathbf{b} \) in the ratio \( m:n \), then the position vector of \( C \) is given by:
\(\mathbf{c} = \frac{n\mathbf{a} + m\mathbf{b}}{m+n}\).
Let's express the problem using this formula. We know:
\[
3\mathbf{a} + 4\mathbf{b} - 7\mathbf{c} = \mathbf{0} \implies 7\mathbf{c} = 3\mathbf{a} + 4\mathbf{b}
\]
The expression \( 7\mathbf{c} = 3\mathbf{a} + 4\mathbf{b} \) suggests that \(\mathbf{c}\) is a weighted average of \(\mathbf{a}\) and \(\mathbf{b}\), as per section formula where:
\[
\mathbf{c} = \frac{4\mathbf{a} + 3\mathbf{b}}{4 + 3}
\]
Hence, \( C \) divides \( AB \) in the ratio \( 4:3 \).
The correct answer is, therefore, \( \text{4:3} \).
By analyzing the derived expression within the context of the vector section formula, it is evident that \( C \) must divide \( AB \) in the ratio \( 4:3 \).
This confirms the provided option.