Linear Combination of Vectors

Q. Two unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are pependicular to each other. Another unit vector $\overrightarrow{c}$ is inclined at an angle $\alpha$ to both $\overrightarrow{a}$ and $\overrightarrow{b}$. If $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}+z(\overrightarrow{a}\times \overrightarrow{b})$, then

1. $x^2+y^2=1$
2. $x^2=y^2$
3. $z^2=-\cos 2\alpha$
4. $x^2+y^2+z^2=1$

Q. Let $\overrightarrow{a}=\hat{i}+\hat{j};\overrightarrow{b}=2\hat{i}-\hat{k}$. Then, vector $\overrightarrow{r}$ satisfying the equations $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{a}$ and $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{b}$ is

1. $\hat{i}-\hat{j}+3\hat{k}$
2. $3\hat{i}-\hat{j}+\hat{k}$
3. $3\hat{i}+\hat{j}-\hat{k}$
4. $\hat{i}-\hat{j}-\hat{k}$

Q. The linear combination of $\overline{a}=\left[\begin{array}{c}1\\ 2\end{array}\right]and\overline{b}=\left[\begin{array}{c}0\\ 3\end{array}\right]$ which gives the vector $\left[\begin{array}{c}2\\ 2\end{array}\right]$ will be ___$\overline{a}$ +__ $\overline{b}$. (If any of the blanks is a fraction, enter the value to the nearest hundredth place).