Let $\bar{a}=\hat{i}+\hat{j}$,$\bar{b}=2\hat{i}-\hat{k}$,and $\bar{c}=3\hat{i}-\hat{j}+\hat{k}$. Find the vector $\bar{p}$ that satisfies $\bar{p} \cdot \bar{a}=0$ and $\bar{p} \times \bar{b}=\bar{c} \times \bar{b}$.

If $a = 2i + 4j - 5k$ and $b = i + 2j + 3k$,then $|a \times b|$ is

Let $\vec{a}=2 \hat{i}-3 \hat{j}+\hat{k}, \vec{b}=\hat{i}+2 \hat{j}-3 \hat{k}, \vec{c}=\hat{i}-\hat{j}$ and $\vec{d}=\hat{i}+\hat{j}+x \hat{k}$. If $(\vec{a} \times \vec{b}) \times \vec{c}$ is perpendicular to $\vec{d}$,then $x=$

$\text{If } \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} - \hat{j} + 3\hat{k} \text{ and } \vec{c} = \hat{i} - \hat{j} \text{ and if } 6\hat{i} + 2\hat{j} + 3\hat{k} = \lambda_1(\vec{a} \times \vec{b}) + \lambda_2(\vec{b} \times \vec{c}) + \lambda_3(\vec{c} \times \vec{a}), \text{ then } (\lambda_1, \lambda_2, \lambda_3) = $

Let $\vec{a}=\hat{i}+\alpha \hat{j}+3 \hat{k}$ and $\vec{b}=3 \hat{i}-\alpha \hat{j}+\hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $8 \sqrt{3}$ square units,then $\vec{a} \cdot \vec{b}$ is equal to ....... .

If $a = i - 2j + 3k$ and $b = 3i + j + 2k,$ then the unit vector perpendicular to $a$ and $b$ is