Question:
The vectors $2\vec{a}+6\vec{b}$ and $3\vec{a}-7\vec{b}$ are position vectors of the points A and B respectively. A point P divides the line segment AB internally in the ratio 3:5. Then $\vec{PB}=$
The vectors $2\vec{a}+6\vec{b}$ and $3\vec{a}-7\vec{b}$ are position vectors of the points A and B respectively. A point P divides the line segment AB internally in the ratio 3:5. Then $\vec{PB}=$
Show Hint
Math Tip: Be careful with the order! "Ratio $m:n$ divides segment $AB$" means $m$ is the multiplier for vector $B$, and $n$ is the multiplier for vector $A$. The vector $\vec{PB}$ is always "destination minus origin", meaning $\vec{OB} - \vec{OP}$.
Updated On: Apr 24, 2026
- $\frac{5\vec{a}-65\vec{b}}{8}$
- $\frac{5\vec{a}-55\vec{b}}{8}$
- $\frac{5\vec{a}-45\vec{b}}{8}$
- $\frac{5\vec{a}-60\vec{b}}{8}$
- $\frac{5\vec{a}+65\vec{b}}{8}$
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Concept:
Vectors - Section Formula.
If point $P$ divides line segment $AB$ internally in the ratio $m:n$, its position vector is $\vec{p} = \frac{m\vec{b} + n\vec{a}}{m + n}$, where $\vec{a}$ and $\vec{b}$ are position vectors of $A$ and $B$.
Step 1: Identify the position vectors and the ratio.
Let origin be $O$. The position vectors are:
Step 2: Apply the section formula to find $\vec{OP}$.
$$ \vec{OP} = \frac{m\vec{OB} + n\vec{OA}}{m + n} $$ $$ \vec{OP} = \frac{3(3\vec{a} - 7\vec{b}) + 5(2\vec{a} + 6\vec{b})}{3 + 5} $$
Step 3: Simplify the position vector of P.
Expand the terms in the numerator: $$ \vec{OP} = \frac{9\vec{a} - 21\vec{b} + 10\vec{a} + 30\vec{b}}{8} $$ Group the $\vec{a}$ and $\vec{b}$ terms together: $$ \vec{OP} = \frac{19\vec{a} + 9\vec{b}}{8} $$
Step 4: Calculate the required vector $\vec{PB}$.
The vector from point $P$ to point $B$ is found by $\vec{PB} = \vec{OB} - \vec{OP}$. $$ \vec{PB} = (3\vec{a} - 7\vec{b}) - \left( \frac{19\vec{a} + 9\vec{b}}{8} \right) $$
Step 5: Find a common denominator and solve.
Multiply the first term by $\frac{8}{8}$ to combine fractions: $$ \vec{PB} = \frac{8(3\vec{a} - 7\vec{b}) - (19\vec{a} + 9\vec{b})}{8} $$ $$ \vec{PB} = \frac{24\vec{a} - 56\vec{b} - 19\vec{a} - 9\vec{b}}{8} $$ Combine like terms: $$ \vec{PB} = \frac{5\vec{a} - 65\vec{b}}{8} $$
Vectors - Section Formula.
If point $P$ divides line segment $AB$ internally in the ratio $m:n$, its position vector is $\vec{p} = \frac{m\vec{b} + n\vec{a}}{m + n}$, where $\vec{a}$ and $\vec{b}$ are position vectors of $A$ and $B$.
Step 1: Identify the position vectors and the ratio.
Let origin be $O$. The position vectors are:
- $\vec{OA} = 2\vec{a} + 6\vec{b}$
- $\vec{OB} = 3\vec{a} - 7\vec{b}$
Step 2: Apply the section formula to find $\vec{OP}$.
$$ \vec{OP} = \frac{m\vec{OB} + n\vec{OA}}{m + n} $$ $$ \vec{OP} = \frac{3(3\vec{a} - 7\vec{b}) + 5(2\vec{a} + 6\vec{b})}{3 + 5} $$
Step 3: Simplify the position vector of P.
Expand the terms in the numerator: $$ \vec{OP} = \frac{9\vec{a} - 21\vec{b} + 10\vec{a} + 30\vec{b}}{8} $$ Group the $\vec{a}$ and $\vec{b}$ terms together: $$ \vec{OP} = \frac{19\vec{a} + 9\vec{b}}{8} $$
Step 4: Calculate the required vector $\vec{PB}$.
The vector from point $P$ to point $B$ is found by $\vec{PB} = \vec{OB} - \vec{OP}$. $$ \vec{PB} = (3\vec{a} - 7\vec{b}) - \left( \frac{19\vec{a} + 9\vec{b}}{8} \right) $$
Step 5: Find a common denominator and solve.
Multiply the first term by $\frac{8}{8}$ to combine fractions: $$ \vec{PB} = \frac{8(3\vec{a} - 7\vec{b}) - (19\vec{a} + 9\vec{b})}{8} $$ $$ \vec{PB} = \frac{24\vec{a} - 56\vec{b} - 19\vec{a} - 9\vec{b}}{8} $$ Combine like terms: $$ \vec{PB} = \frac{5\vec{a} - 65\vec{b}}{8} $$
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Vector basics Questions
The projection of the vector \(a=2i-3j+4k \)on the vector \(b=i+2j+2k\) is ?
- If $2\hat{i} - \hat{j} + \hat{k} = s(3\hat{i} - 4\hat{j} - 4\hat{k}) + t(\hat{i} - 3\hat{j} - 5\hat{k})$, where $s$ and $t$ are scalars, then $3s + 5t$ is equal to:
- The position vectors of the vertices of a triangle are $\hat{i}+2\hat{j}-4\hat{k}$, $-\hat{i}-2\hat{j}-4\hat{k}$ and $2\hat{i}+3\hat{j}+5\hat{k}$. The perimeter of the triangle is
- Let $|\vec{a}|=2,|\vec{b}|=\sqrt{13+6\sqrt{3}}$ and $|\vec{c}|=3$. If $\vec{a}-\vec{b}+\vec{c}=0$, then the angle between $\vec{a}$ and $\vec{c}$ is
- Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $|\vec{a}|=2, |\vec{b}|=3, |\vec{c}|=4$ and $\vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c}=0$. If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$, then $\vec{a}$ is equal to
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions