Represent and use a scalar multiple of a vector.

The vector \ (\vec{u} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}\) is shown on the diagram below. Which of the following diagrams correctly represents the vector \ (3\vec{u}\)?

Given vectors $\mathbf{u} = 2\mathbf{i} - 3\mathbf{j} + 5\mathbf{k}$ and $\mathbf{v} = -\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}$, determine the value of the scalar $m$ such that the vector $\mathbf{w} = m\mathbf{u} + 3\mathbf{v}$ is perpendicular to $\mathbf{u}$.

A helicopter departs from point P and travels with displacement vector $\vec{d} = 3\vec{i} + 4\vec{j} + 2\vec{k}$ (in kilometres) to reach point Q. The helicopter then continues from Q in the same direction for a total displacement from P that is 2.5 times the original displacement. (a) Determine the position vector of the helicopter at its final location R, relative to point P. (1 mark) (b) Calculate the total distance travelled by the helicopter from P to R. (1 mark) (c) Express the vector $\overrightarrow{QR}$ in component form. (1 mark)