Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}+\vec{b}|=15$ and

$$ \vec{a} \times(3 \hat{i}-4 \hat{j}+5 \hat{k})=(3 \hat{i}-4 \hat{j}+5 \hat{k}) \times \vec{b} $$

What is the value of $|(\vec{a}+\vec{b}) \cdot(2 \hat{i}+3 \hat{j}+\hat{k})|$ ?

Let $S$ be the set of all unit vectors in the $X Y$-plane. Then the set $S$ has

Consider the vectors $\vec{a}=\hat{\imath}+x \hat{\jmath}+2 \hat{k}, \vec{b}=\hat{\imath}+2 \hat{\jmath}+x \hat{k}, \vec{c}=2 \hat{\imath}+\hat{\jmath}+3 \hat{k}$. The values of $x$ for Which there is at least one nonzero vector perpendicular to the vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are