If $|\bar{a}|=\sqrt{26}$,$|\bar{b}|=7$,and $|\bar{a} \times \bar{b}|=35$,then $\bar{a} \cdot \bar{b}=$

If $A, B, C$ and $D$ are points whose position vectors are $\hat{i}+\hat{j}+\hat{k}, 4 \hat{i}-\hat{j}+2 \hat{k}, 5 \hat{i}+\hat{j}$ and $7 \hat{i}+2 \hat{j}+3 \hat{k}$ respectively,then the projection of $\vec{AB}$ on $\vec{CD}$ is

The vector $\overline{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}$ lies in the plane of the vectors $\bar{b}=\hat{i}+\hat{j}$ and $\bar{c}=\hat{j}+\hat{k}$ and bisects the angle between $\bar{b}$ and $\bar{c}$. Then which one of the following gives possible values of $\alpha$ and $\beta$?

If $a, b$ and $c$ are perpendicular to $b + c, c + a$ and $a + b$ respectively,and if $|a + b| = 6, |b + c| = 8$ and $|c + a| = 10$,then $|a + b + c| = $

Let $x \in R$ and $\log_2 x > 0$. Then,the vectors $A = (2, \log_2 x, s)$ and $B = (\log_2 x, s, \log_2 x)$ include an acute angle if

If $\vec{a}=2 \hat{i}+\hat{j}-\hat{k}$,$\vec{b}=\hat{i}-\hat{j}+3 \hat{k}$,$\vec{x}=\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right) \vec{b}$,$\vec{y}=\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}\right) \vec{a}$ and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$,then $x^2+y^2=$