The point of intersection of the lines
\[
\vec{r} = 2\vec{b} + t(6\vec{c} - \vec{a}) \quad \text{and} \quad \vec{r} = \vec{a} + s(\vec{b} - 3\vec{c})
\text{ is:}
\]
Show Hint
- $\dfrac{\vec{a} + \vec{b} + \vec{c}}{3}$
- $\vec{b} - \vec{c} - 6\vec{a}$
- $2\vec{a} - \vec{b} + \vec{c}$
- $\vec{a} + 2\vec{b} - 6\vec{c}$
The Correct Option is D
Solution and Explanation
Top Questions on Vectors
- Consider the vectors $$ \vec{x} = \hat{i} + 2\hat{j} + 3\hat{k},\quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},\quad \vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}. $$ For two distinct positive real numbers $ \alpha $ and $ \beta $, define $$ \vec{X} = \alpha \vec{x} + \beta \vec{y} - \vec{z},\quad \vec{Y} = \alpha \vec{y} + \beta \vec{z} - \vec{x},\quad \vec{Z} = \alpha \vec{z} + \beta \vec{x} - \vec{y}. $$ If the vectors $ \vec{X}, \vec{Y}, \vec{Z} $ lie in a plane, then the value of $ \alpha + \beta - 3 $ is ________.
Let $ \vec{w} = \hat{i} + \hat{j} - 2\hat{k} $, and $ \vec{u} $ and $ \vec{v} $ be two vectors, such that $ \vec{u} \times \vec{v} = \vec{w} $ and $ \vec{v} \times \vec{w} = \vec{u} $. Let $ \alpha, \beta, \gamma $, and $ t $ be real numbers such that: $$ \vec{u} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}, $$ and the system of equations is: $$ -t\alpha + \beta + \gamma = 0 \quad \cdots (1) $$ $$ \alpha - t\beta + \gamma = 0 \quad \cdots (2) $$ $$ \alpha + \beta - t\gamma = 0 \quad \cdots (3) $$ Match each entry in List-I to the correct entry in List-II and choose the correct option.
List-I- [(P)] $ |\vec{v}|^2 $ is equal to
- [(Q)] If $ \alpha = \sqrt{3} $, then $ \gamma^2 $ is equal to
- [(R)] If $ \alpha = \sqrt{3} $, then $ (\beta + \gamma)^2 $ is equal to
- [(S)] If $ \alpha = \sqrt{2} $, then $ t + 3 $ is equal to
List-II
- [(1)] 0
- [(2)] 1
- [(3)] 2
- [(4)] 3
- [(5)]
- Find the shortest distance between the lines: \[ \frac{x+1}{2} = \frac{y-1}{1} = \frac{z-9}{-3} \] and \[ \frac{x-3}{2} = \frac{y+15}{-7} = \frac{z-9}{5}. \]
Let \( \vec{p} \) and \( \vec{q} \) be two unit vectors and \( \alpha \) be the angle between them. Then \( (\vec{p} + \vec{q}) \) will be a unit vector for what value of \( \alpha \)?
- Let \( \vec{p} = 2\hat{i} - 3\hat{j} - \hat{k} \), \( \vec{q} = -3\hat{i} + 4\hat{j} + \hat{k} \) and \( \vec{r} = \hat{i} + \hat{j} + 2\hat{k} \). Express \( \vec{r} \) in the form of \( \vec{r} = \lambda \vec{p} + \mu \vec{q} \) and hence find the values of \( \lambda \) and \( \mu \).
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