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The position vector of the point which d...

The position vector of the point which divides the join of points `2vec(a)-3vec(b)` and `vec(a)+vec(b)` in the ratio `3:1` is

A

`(3vec(a)-2vec(b))/(2)`

B

`(7vec(a)-8vec(b))/(4)`

C

`(3vec(a))/(4)`

D

`(5vec(a))/(4)`

Text Solution

AI Generated Solution

The correct Answer is:
To find the position vector of the point that divides the line segment joining the points \(2\vec{a} - 3\vec{b}\) and \(\vec{a} + \vec{b}\) in the ratio \(3:1\), we can use the section formula for vectors. ### Step-by-step Solution: 1. **Identify the Points and the Ratio**: Let \(A = 2\vec{a} - 3\vec{b}\) and \(B = \vec{a} + \vec{b}\). The ratio in which the point divides the segment \(AB\) is \(3:1\). 2. **Use the Section Formula**: The position vector \( \vec{C} \) of the point \(C\) that divides the line segment joining points \(A\) and \(B\) in the ratio \(m:n\) is given by: \[ \vec{C} = \frac{m\vec{B} + n\vec{A}}{m+n} \] Here, \(m = 3\) and \(n = 1\). 3. **Substituting the Values**: Substitute \(A\) and \(B\) into the formula: \[ \vec{C} = \frac{3(\vec{a} + \vec{b}) + 1(2\vec{a} - 3\vec{b})}{3 + 1} \] 4. **Simplifying the Expression**: Calculate the numerator: \[ 3(\vec{a} + \vec{b}) = 3\vec{a} + 3\vec{b} \] \[ 1(2\vec{a} - 3\vec{b}) = 2\vec{a} - 3\vec{b} \] Now combine these: \[ 3\vec{a} + 3\vec{b} + 2\vec{a} - 3\vec{b} = (3\vec{a} + 2\vec{a}) + (3\vec{b} - 3\vec{b}) = 5\vec{a} \] 5. **Final Calculation**: Now divide by \(m+n = 4\): \[ \vec{C} = \frac{5\vec{a}}{4} \] Thus, the position vector of the point that divides the line segment in the ratio \(3:1\) is: \[ \vec{C} = \frac{5\vec{a}}{4} \]
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