Define and use multiplication by a scalar of a vector in Cartesian form.

Given the vector $\mathbf{u} = 3\mathbf{i} - 5\mathbf{j} + 2\mathbf{k}$, determine $-4\mathbf{u}$.

Three vectors are defined by their components in the table below. Calculate the vector \(2\mathbf{u} + 3\mathbf{v}\), expressing your answer in the form \(a\mathbf{i} + b\mathbf{j} + c\mathbf{k}\).

A telecommunications tower is designed with three support cables. Cable A runs from the top of the tower to ground level, and is represented by the vector **a** = 12**i** + 5**j** − 8**k**, where distances are measured in metres. Cable B is designed to be parallel to cable A but extend to a point that is 2.5 times as far from the tower base. (a) Determine the vector **b** that represents cable B. (b) Use your result from part (a) to determine the horizontal distance from the tower base to the anchor point of cable B. Show evidence of the calculations used to find this distance.

Given the vector **a** = 3**i** − 2**j** + 5**k** shown in the diagram, determine the vector −2**a**.

Consider the vector **v** = 3**i** − 5**j** + 2**k** and the scalar *m* = −4. (a) Determine the vector *m***v** in Cartesian form. (b) Use your result from part (a) to determine the magnitude of *m***v**, expressing your answer in simplest surd form.