If $\overrightarrow{a}=\hat{i}-\hat{j}-\hat{k}$ and $\overrightarrow{b}=\lambda \hat{i}-3 \hat{j}+\hat{k}$ and the orthogonal projection of $\overrightarrow{b}$ on $\overrightarrow{a}$ is $\frac{4}{3}(\hat{i}-\hat{j}-\hat{k})$,then $\lambda$ is equal to

Let $\overline{A}, \overline{B}, \overline{C}$ be vectors of lengths $3$ units,$4$ units,and $5$ units respectively. If $\overline{A}$ is perpendicular to $\overline{B}+\overline{C}$,$\overline{B}$ is perpendicular to $\overline{C}+\overline{A}$,and $\overline{C}$ is perpendicular to $\overline{A}+\overline{B}$,then the length of vector $\overline{A}+\overline{B}+\overline{C}$ is

$A$ vector coplanar with vectors $i + j$ and $j + k$ and parallel to the vector $2i - 2j - 4k$ is:

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \overrightarrow{a} + 2 \overrightarrow{b}$ and $3 \overrightarrow{a} - \lambda \overrightarrow{b}$ are perpendicular to each other,then the number of values of $\lambda$ in $[-1, 3]$ is:

The component of $\vec{i} + \vec{j}$ along $\vec{j} + \vec{k}$ is:

Let $\vec{r}$ be a vector in the plane of $\hat{i} - 2\hat{j} + \hat{k}$ and $\hat{i} - \hat{j} - \hat{k}$ such that $\vec{r} \cdot (\hat{i} + \hat{j}) + 2 = 0$ and the length of the projection of $\vec{r}$ on $\hat{i} - \hat{j}$ is $4\sqrt{2}$. Then,the magnitude of vector $\vec{r}$ is: