If $\overrightarrow{a}=3\hat{i}+\hat{j}+2\hat{k}$ and $\overrightarrow{b}=2\hat{i}-2\hat{j}+4\hat{k}$, then find a unit vector perpendicular to both the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.

Find a vector of magnitude $6$ units which is perpendicular to both the vectors $2\hat{i}-\hat{j}+2\hat{k}$ and $4\hat{i}-\hat{j}+3\hat{k}$.

Find a vector of magnitude $19$ which is perpendicular to both the vectors $-\hat{j}+\hat{k}$ and $4\hat{i}-\hat{j}+8\hat{k}$.

Find a vector of magnitude $6$, perpendicular to each of the vectors $\left(\overrightarrow{a}+\overrightarrow{b}\right)$ and $\left(\overrightarrow{a}-\overrightarrow{b}\right)$ where $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}+3\hat{k}$.

If $\overrightarrow{a}=\hat{i}-2\hat{j}+3\hat{k}$ and $\overrightarrow{b}=2\hat{i}+3\hat{j}-5\hat{k}$, find $\overrightarrow{a}\times \overrightarrow{b}$. Verify that $\overrightarrow{a}$ and $\overrightarrow{a}\times \overrightarrow{b}$ are perpendicular to each other.

(Answer in the form $a\hat{i}+b\hat{j}+c\hat{k}$)

If $\overrightarrow{a}=\hat{i}-\hat{j},$ $\overrightarrow{b}=3\hat{i}-\hat{k}$ and $\overrightarrow{c}=7\hat{i}-\hat{k}$, find a vector $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$ and $\overrightarrow{c}·\overrightarrow{d}=1$

(If Answer is in the form of $a\hat{i}+b\hat{j}+c\hat{k}$,then find the value of $4\left(a+b+c\right)$)

If $\overrightarrow{a}=\hat{i}+4\hat{j}+2\hat{k}$, $\overrightarrow{b}=3\hat{i}-2\hat{j}+7\hat{k}$ and $\overrightarrow{c}=2\hat{i}-\hat{j}+4\hat{k}$, find a vector $\overrightarrow{d}$ which is perpendicular to both the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$and $\overrightarrow{c}·\overrightarrow{d}=27$.

(Answer in the form $a\hat{i}+b\hat{j}+c\hat{k}$)

Let $\overrightarrow{a}=\hat{i}+4\hat{j}+2\hat{k}$, $\overrightarrow{b}=3\hat{i}-2\hat{j}+7\hat{k}$  and $\overrightarrow{c}=2\hat{i}-\hat{j}+4\hat{k}$. Find a vector $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$ and $\overrightarrow{c}·\overrightarrow{d}=18.$

(Answer in the form $a\hat{i}+b\hat{j}+c\hat{k}$)

If $\overrightarrow{a}=4\hat{i}+5\hat{j}-\hat{k}$, $\overrightarrow{b}=\hat{i}-4\hat{j}+5\hat{k}$ and $\overrightarrow{c}=3\hat{i}+\hat{j}-\hat{k}$, find a vector $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{c}$ and $\overrightarrow{b}$ and $\overrightarrow{d}·\overrightarrow{a}=21.$

(Answer in the form $a\hat{i}+b\hat{j}+c\hat{k}$)