The sine of the angle between vectors $$a = 2\widehat i - 6\widehat j - 3\widehat k$$ and $$b = 4\widehat i + 3\widehat j - \widehat k$$ is

What is the value of $$\lambda$$ for which the vectors $$3\widehat i + 4\widehat j - \widehat k$$ and $$ - 2\widehat i + \lambda \widehat j + 10\widehat k$$ are perpendicular?

Let $\vec{p}=\vec{a}-\vec{b},\vec{q}=\vec{a}+\vec{b}$. If $|\vec{a}|=|\vec{b}|=2$ and $\vec{a}\cdot \vec{b}=2$, then what is the value of $|\vec{p}\times \vec{q}|$?

How many of the following can be a vector perpendicular to both the vectors $2\hat{i} - \hat{j} + \hat{k}$ and $\hat{i} + \hat{j} + 3\hat{k}$ ?

I. $4\hat{i}+5\hat{j}-3\hat{k}$

II. $-8\hat{i}-10\hat{j}+6\hat{k}$

III. $\frac{1}{50}(-4\hat{i}-5\hat{j}+3\hat{k})$

Select the correct answer.