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 $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{j}-\hat{k},$ find a vector $\vec{c}$ such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3.$

Let $\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}$ and $\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k} .$ Find a vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b},$ and $\vec{c} \cdot \vec{d}=15.$

Let $\overrightarrow{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{b}=\lambda \hat{j}+2 \hat{k}$, where $\lambda \in \mathbb{Z}$, be two vectors. Let $\overrightarrow{c}=\overrightarrow{a} \times \overrightarrow{b}$ and $\overrightarrow{d}$ be a vector of magnitude $2$ in the $yz$-plane. If $|\overrightarrow{c}|=\sqrt{53}$, then the maximum possible value of $(\overrightarrow{c} \cdot \overrightarrow{d})^2$ is equal to:

Find $|\vec{a} \times \vec{b}|,$ if $\vec{a}=\hat{i}-7 \hat{j}+7 \hat{k}$ and $\vec{b}=3 \hat{i}-2 \hat{j}+2 \hat{k}.$

$a, b, c, d$ are coplanar vectors, then $(a \times b) \times (c \times d)$ is equal to