If $\overline{a}$  and  $\overline{b}$ be two vectors such tha t   $\overline{a}=2\overline{i}+2\overline{j}+p\overline{k}\left|\overline{b}\right|=7,\overline{a}.\overline{b}=4$  and  $\left|\overline{a}\times \overline{b}\right|=5\sqrt{17}$ then p =

$\begin{array}{rl} & \text{Given}\overline{a}=2\overline{i}+2\overline{j}+p\overline{k}\Rightarrow \left|\overline{a}\right|=\sqrt{8+p^2}\text{and}\left|\overline{b}\right|=7\\ & \overline{a}\cdot \overline{b}=4\Rightarrow \left|\overline{a}\right|\left|\overline{b}\right|\cos \theta =4\\ & |\overline{a}\times \overline{b}|=5\sqrt{17}\Rightarrow \left|\overline{a}\right|\left|\overline{b}\right|\sin \theta =5\sqrt{17}\end{array}$
$\begin{array}{rl} & \Rightarrow \frac{4}{\cos \theta }\times \sin \theta =5\sqrt{17}\\ & \Rightarrow \tan \theta =\frac{5\sqrt{17}}{4}\\ & \Rightarrow \cos \theta =\frac{4}{21}\\ & \Rightarrow \frac{4}{\left|\overline{a}\right|\left|\overline{b}\right|}=\frac{4}{21}\\ & \Rightarrow \left|\overline{a}\right|\left|\overline{b}\right|=21\\ & \Rightarrow \sqrt{8+p^2}\times 7=21\\ & \Rightarrow 8+p^2=9\\ & \Rightarrow p^2=1\\ & \Rightarrow p=\pm 1\end{array}$