Let $\quad \bar{a}=\alpha \hat{i}+3 \hat{j}-\hat{k}, \bar{b}=3 \hat{i}-\hat{j}+\beta \hat{k} \quad$ and $\bar{c}=\hat{i}+2 \hat{j}-2 \hat{k}$ where $\alpha, \beta \in \mathbb{R}$, be three vectors. If the projection of $\bar{a}$ on $\bar{c}$ is $\frac{10}{3}$ and $\overline{\mathrm{b}} \times \overline{\mathrm{c}}=-6 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}$, then the value of $(\alpha+\beta)$ is equal to

If $\bar{a}, \bar{b}, \bar{c}$ are three coplanar vectors such that $|\overline{\mathrm{a}}|=1,|\overline{\mathrm{~b}}|=2, \overline{\mathrm{~b}} \cdot \overline{\mathrm{c}}=8$, the angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ is $45^{\circ}$, then $|\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})|=$

Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be vectors of magnitude 2,3 and 4 respectively. If $\bar{a}$ is perpendicular to $(\overline{\mathrm{b}}+\overline{\mathrm{c}}), \overline{\mathrm{b}}$ is perpendicular to ( $\overline{\mathrm{c}}+\overline{\mathrm{a}}$ ) and $\overline{\mathrm{c}}$ is perpendicular to $(\bar{a}+\bar{b})$, then the magnitude of $\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}$ is

If $\quad \overline{\mathrm{a}}=\lambda x \hat{\mathrm{i}}+y \hat{\mathrm{j}}+4 z \hat{\mathrm{k}}, \quad \overline{\mathrm{b}}=y \hat{\mathrm{i}}+x \hat{\mathrm{j}}+3 y \hat{\mathrm{k}}$, $\overline{\mathrm{c}}=-z \hat{\mathrm{i}}-2 z \hat{\mathrm{j}}-(\lambda+1) \hat{\mathrm{k}} x$ are the sides of the triangle ABC , where $x, y, \mathrm{z}$ are not all zero, such that $\bar{a}+\bar{b}-\bar{c}=\overline{0}$, then value of $\lambda$ is