If \( \mathbf{a} \) and \( \mathbf{b} \) are position vectors of two points \( P \) and \( Q \) respectively, then find the position vector of a point \( R \) in \( QP \) produced such that
\[
QR = \frac{3}{2} QP.
\]
Show Hint
Solution and Explanation
Let the position vectors of \( P \) and \( Q \) be: \[ \mathbf{OP} = \mathbf{a}, \quad \mathbf{OQ} = \mathbf{b}. \] The vector \( QP \) is given by: \[ \mathbf{QP} = \mathbf{a} - \mathbf{b}. \] Step 2: Express \( QR \) in terms of \( QP \).
Given that: \[ QR = \frac{3}{2} QP, \] we write: \[ \mathbf{QR} = \frac{3}{2} (\mathbf{a} - \mathbf{b}). \] Step 3: Compute the position vector of \( R \).
Using the relation: \[ \mathbf{OR} = \mathbf{OQ} + \mathbf{QR}, \] \[ \mathbf{OR} = \mathbf{b} + \frac{3}{2} (\mathbf{a} - \mathbf{b}). \] Expanding: \[ \mathbf{OR} = \mathbf{b} + \frac{3}{2} \mathbf{a} - \frac{3}{2} \mathbf{b}. \] \[ \mathbf{OR} = \frac{3}{2} \mathbf{a} + \left(1 - \frac{3}{2} \right) \mathbf{b}. \] \[ \mathbf{OR} = \frac{3}{2} \mathbf{a} - \frac{1}{2} \mathbf{b}. \] Final Answer: \[ \mathbf{r} = \frac{3}{2} \mathbf{a} - \frac{1}{2} \mathbf{b}. \]
Top Questions on Vectors
Let \( \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} \), \( \vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k} \) and a vector \( \vec{c} \) be such that \[ (\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k} \] and \[ \vec{a} \cdot \vec{c} = 3. \] If \( \vec{b} \times \vec{c} = \vec{d} \), then find \( |\vec{a} \cdot \vec{d}| \).
- If the equation of the line passing through the point $ \left( 0, -\frac{1}{2}, 0 \right) $ and perpendicular to the lines $ \mathbf{r_1} = \lambda ( \hat{i} + a \hat{j} + b \hat{k}) \quad \text{and} \quad \mathbf{r_2} = ( \hat{i} - \hat{j} - 6 \hat{k} ) + \mu( -b \hat{i} + a \hat{j} + 5 \hat{k}), $ is $ \frac{x - 1}{-2} = \frac{y + 4}{d} = \frac{z - c}{-4}, $ then $ a + b + c + d $ is equal to:
Let $ \vec{a} = \hat{i} + 2\hat{j} + \hat{k} $, $ \vec{b} = 3\hat{i} - 3\hat{j} + 3\hat{k} $, $ \vec{c} = 2\hat{i} - \hat{j} + 2\hat{k} $ and $ \vec{d} $ be a vector such that $ \vec{b} \times \vec{d} = \vec{c} \times \vec{d} $ and $ \vec{a} \cdot \vec{d} = 4 $. Then $ |\vec{a} \times \vec{d}|^2 $ is equal to _______
Consider two vectors $\vec{u} = 3\hat{i} - \hat{j}$ and $\vec{v} = 2\hat{i} + \hat{j} - \lambda \hat{k}$, $\lambda>0$. The angle between them is given by $\cos^{-1} \left( \frac{\sqrt{5}}{2\sqrt{7}} \right)$. Let $\vec{v} = \vec{v}_1 + \vec{v}_2$, where $\vec{v}_1$ is parallel to $\vec{u}$ and $\vec{v}_2$ is perpendicular to $\vec{u}$. Then the value $|\vec{v}_1|^2 + |\vec{v}_2|^2$ is equal to
- Let $ \vec{a} = \hat{i} + \hat{j} + \hat{k} $, $ \vec{b} = 3\hat{i} + 2\hat{j} - \hat{k} $, $ \vec{c} = \lambda \hat{j} + \mu \hat{k} $ and $ \hat{d} $ be a unit vector such that $ \vec{a} \times \hat{d} = \vec{b} \times \hat{d} $ and $ \vec{c} \cdot \hat{d} = 1 $. If $ \vec{c} $ is perpendicular to $ \vec{a} $, then $ |3\lambda \hat{d} + \mu \vec{c}|^2 $ is equal to ____.
Questions Asked in CBSE CLASS XII exam
- Consider the Linear Programming Problem, where the objective function \[ Z = x + 4y \] needs to be minimized subject to the following constraints: \[ 2x + y \geq 1000, \] \[ x + 2y \geq 800, \] \[ x \geq 0, \quad y \geq 0. \] Draw a neat graph of the feasible region and find the minimum value of $Z$.
- CBSE CLASS XII - 2025
- Linear Programming Problem
You are Mr. Abhishek Sharma. You are planning to celebrate your parents’ 50th wedding anniversary in a grand manner. Draft a formal invitation card for your friends and relatives. Mention all necessary details.
- CBSE CLASS XII - 2025
- Writing Skills
- Draw the energy-band diagram for conductors, semiconductors, and insulators at T = 0 K. How is an electron-hole pair formed in a semiconductor at room temperature?
- CBSE CLASS XII - 2025
- Solid State Physics, Devices, Electronics
- Inside a nucleus, the nuclear forces between proton and proton, proton and neutron, neutron and neutron are \( F_{pp} \), \( F_{pn} \), and \( F_{nn} \) respectively. Then:
- CBSE CLASS XII - 2025
- Nuclear physics
- Explain the beneficial role of the following, produced as a result of the processes of biotechnology, to mankind:
(a) Cow named Rosie
(b) \(\alpha\)-1-antitrypsin- CBSE CLASS XII - 2025
- Applications of Biotechnology