If $\overrightarrow{r}=\hat{i}+\hat{j}+\lambda (2\hat{i}+\hat{j}+4\hat{k})$  and $\overrightarrow{r}·(\hat{i}+2\hat{j}-\hat{k})=3$ are the equations of a line and a plane respectively, then which of the following is incorrect?

If  vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ are two adjacent sides of a parallelogram, then the vector representing the altitude of the parallelogram which is perpendicular to $\overrightarrow{\mathrm{a}}$ is

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are non-coplanar unit vectors also $\overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are  non-collinear and $2\overrightarrow{\mathrm{a}}\times \left(\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}\right)=\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}$, then

If  $\overrightarrow{\mathrm{a}}=\frac{1}{7}\left(2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+6\hat{\mathrm{k}}\right); \overrightarrow{\mathrm{b}}=\frac{1}{7}\left(6\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}}\right); \overrightarrow{\mathrm{c}}=c_1\hat{\mathrm{i}}+c_2\hat{\mathrm{j}}+c_3\hat{\mathrm{k}}$, matrix $A=\left[\begin{array}{ccc}\frac{2}{7}& \frac{3}{7}& \frac{6}{7}\\ \frac{6}{7}& \frac{2}{7}& \frac{-3}{7}\\ c_1& c_2& c_3\end{array}\right]$ and $A{A}^T=I,$then $\overrightarrow{\mathrm{c}}$ is 

Prove Cauchy-Schwarz inequality: $(\overrightarrow{\mathrm{a}}·\overrightarrow{\mathrm{b}}{)}^2\le {\left|\overrightarrow{\mathrm{a}}\right|}^2·{\left|\overrightarrow{\mathrm{b}}\right|}^2$.