If $\bar{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}, \quad \bar{b}=-\hat{i}+2 \hat{j}+\hat{k}$ and $\bar{c}=3 \hat{i}+\hat{j}$ are the vectors such that $\overline{\mathrm{a}}+\lambda \overline{\mathrm{b}}$ is perpendicular to $\bar{c}$, then value of $\lambda$ is

If $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are three vectors such that $\overline{\mathrm{a}} \neq \overline{0}$ and $\overline{\mathrm{a}} \times \overline{\mathrm{b}}=2 \overline{\mathrm{a}} \times \overline{\mathrm{c}},|\overline{\mathrm{a}}|=|\overline{\mathrm{c}}|=1,|\overline{\mathrm{~b}}|=4$ and $|\overline{\mathrm{b}} \times \overline{\mathrm{c}}|=\sqrt{15}$. If $\overline{\mathrm{b}}-2 \overline{\mathrm{c}}=\lambda \overline{\mathrm{a}}$, then $\lambda$ is

If $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ are unit coplanar vectors, then the scalar triple product $\left[\begin{array}{lll}2 \overline{\mathrm{a}}-\overline{\mathrm{b}} & 2 \overline{\mathrm{~b}}-\overline{\mathrm{c}} & 2 \overline{\mathrm{c}}-\overline{\mathrm{a}}\end{array}\right]$ has the value

Let the vectors $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ be such that $|\bar{a}|=2,|\bar{b}|=4$ and $|\bar{c}|=4$. If the projection of $\bar{b}$ on $\bar{a}$ is equal to the projection of $\bar{c}$ on $\bar{a}$ and $\bar{b}$ is perpendicular to $\bar{c}$, then the value of $|\vec{a}+\bar{b}-\bar{c}|$ is