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
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
Show 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 $\mathbf{a}$).
Updated On: May 2, 2026
- 4:3
- 4:7
- 3:7
- 3:4
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
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 $4:3$.
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 $4:3$.
Was this answer helpful?
0
0
Top MET Mathematics Questions
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- The equation $3^{3x + 4} = 9^{2x - 2},\ x>0$ has the solution
- A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is
- Every group of order 7 is
- Let $U_{n} = 2 + 2^{3} + 2^{5} + \cdots + 2^{2n+1}$ and $V_{n} = 1 + 4 + 4^{2} + \cdots + 4^{n-1}$. Then $\displaystyle\lim_{n \to \infty} \dfrac{U_n}{V_n}$ is equal to
View More Questions
Top MET 3D Geometry Questions
- A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is
- If $\left(\dfrac{1}{2},\,\dfrac{1}{3},\,n\right)$ are the direction cosines of a line, then the value of $n$ is
- If $\mathbf{a}= -\hat{i} + 2\hat{j} - \hat{k}$, $\mathbf{b} = \hat{i} + \hat{j} - 3\hat{k}$ and $\mathbf{c} = -4\hat{i} - \hat{k}$, then $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$ is
- The work done by the force $4\hat{i} - 3\hat{j} + 2\hat{k}$ in moving a particle along a straight line from the point $(3, 2, -1)$ to $(2, -1, 4)$ is
- If $|\mathbf{a} + \mathbf{b}| = |\mathbf{a} - \mathbf{b}|$, then $\mathbf{a}$ and $\mathbf{b}$ are
View More Questions
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $
View More Questions