The vector in the direction of the vector $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ that has magnitude 9 is

The position vector of the point which divides the join of points $2 \overrightarrow{\mathbf{a}}-3 \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}$ in the ratio $3: 1$, is

The vector having initial and terminal points as $(2,5,0)$ and $(-3,7$, 4), respectively is

The angle between two vectors $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ with magnitudes $\sqrt{3}$ and 4, respectively and $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=2 \sqrt{3}$ is

Find the value of $\lambda$ such that the vectors $\overrightarrow{\mathbf{a}}=2 \hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ are orthogonal.