Let $\vec{a} = xi + yj + zk$ and $x = 2y$. If $|\vec{a}| = 5\sqrt{2}$ and $\vec{a}$ makes an angle of $135^\circ$ with the z-axis then $\vec{a} =$
Show Hint
- $2\sqrt{3}i + \sqrt{3}j - 3k$
- $2\sqrt{6}i + \sqrt{6}j - 6k$
- $2\sqrt{5}i + \sqrt{5}j - 5k$
- $2\sqrt{5}i + \sqrt{5}j + 5k$
The Correct Option is C
Solution and Explanation
Given $|\vec{a}| = 5\sqrt{2}$, we have: $\sqrt{(2y)^2 + y^2 + z^2} = 5\sqrt{2}$
$\Rightarrow \sqrt{5y^2 + z^2} = 5\sqrt{2} \Rightarrow 5y^2 + z^2 = 50$
Also, $\cos(135^\circ) = \frac{z}{|\vec{a}|} = -\frac{1}{\sqrt{2}} \Rightarrow z = -5$
Substituting in the equation: $5y^2 + 25 = 50 \Rightarrow y^2 = 5 \Rightarrow y = \sqrt{5}$
Then $x = 2\sqrt{5}$ and $\vec{a} = 2\sqrt{5}i + \sqrt{5}j - 5k$
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 \( \mu \).
Questions Asked in AP EAPCET exam
- A car accelerates uniformly from 10 m/s to 30 m/s in 10 seconds. What is the distance covered?
- AP EAPCET - 2025
- Kinematics
- Which blood component is responsible for clotting?
- AP EAPCET - 2025
- Human physiology
- Which element has the smallest atomic radius among the following?
- AP EAPCET - 2025
- Atomic Structure
- A ball is thrown vertically upward with a speed of 20 m/s. How high will it rise before coming to rest momentarily? (Take g = 10 m/s2)
- AP EAPCET - 2025
- Kinematics
- Which plant hormone is responsible for apical dominance?
- AP EAPCET - 2025
- Plant Physiology