1
JEE Advanced 2024 Paper 2 Online
Numerical
+4
-0

Let $\vec{p}=2 \hat{i}+\hat{j}+3 \hat{k}$ and $\vec{q}=\hat{i}-\hat{j}+\hat{k}$. If for some real numbers $\alpha, \beta$, and $\gamma$, we have

$$15 \hat{i}+10 \hat{j}+6 \hat{k}=\alpha(2 \vec{p}+\vec{q})+\beta(\vec{p}-2 \vec{q})+\gamma(\vec{p} \times \vec{q}),$$

then the value of $\gamma$ is ________.

2
JEE Advanced 2024 Paper 1 Online
Numerical
+4
-0

Let $\overrightarrow{O P}=\frac{\alpha-1}{\alpha} \hat{i}+\hat{j}+\hat{k}, \overrightarrow{O Q}=\hat{i}+\frac{\beta-1}{\beta} \hat{j}+\hat{k}$ and $\overrightarrow{O R}=\hat{i}+\hat{j}+\frac{1}{2} \hat{k}$ be three vectors, where $\alpha, \beta \in \mathbb{R}-\{0\}$ and $O$ denotes the origin. If $(\overrightarrow{O P} \times \overrightarrow{O Q}) \cdot \overrightarrow{O R}=0$ and the point $(\alpha, \beta, 2)$ lies on the plane $3 x+3 y-z+l=0$, then the value of $l$ is ____________.

3
JEE Advanced 2023 Paper 1 Online
Numerical
+4
-0
Let $P$ be the plane $\sqrt{3} x+2 y+3 z=16$ and let $S=\left\{\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}: \alpha^2+\beta^2+\gamma^2=1\right.$ and the distance of $(\alpha, \beta, \gamma)$ from the plane $P$ is $\left.\frac{7}{2}\right\}$. Let $\vec{u}, \vec{v}$ and $\vec{w}$ be three distinct vectors in $S$ such that $|\vec{u}-\vec{v}|=|\vec{v}-\vec{w}|=|\vec{w}-\vec{u}|$. Let $V$ be the volume of the parallelepiped determined by vectors $\vec{u}, \vec{v}$ and $\vec{w}$. Then the value of $\frac{80}{\sqrt{3}} V$ is :
4
JEE Advanced 2021 Paper 1 Online
Numerical
+4
-0
Let $$\overrightarrow u$$, $$\overrightarrow v$$ and $$\overrightarrow w$$ be vectors in three-dimensional space, where $$\overrightarrow u$$ and $$\overrightarrow v$$ are unit vectors which are not perpendicular to each other and $$\overrightarrow u$$ . $$\overrightarrow w$$ = 1, $$\overrightarrow v$$ . $$\overrightarrow w$$ = 1, $$\overrightarrow w$$ . $$\overrightarrow w$$ = 4

If the volume of the paralleopiped, whose adjacent sides are represented by the vectors, $$\overrightarrow u$$, $$\overrightarrow v$$ and $$\overrightarrow w$$, is $$\sqrt 2$$, then the value of $$\left| {3\overrightarrow u + 5\overrightarrow v } \right|$$ is ___________.