Use multiplication by a scalar of a vector in Cartesian form.

Calculate \(2\vec{v}\) where \(\vec{v} = 3\mathbf{i} - 5\mathbf{j} + 4\mathbf{k}\).

A drone delivery service models the displacement of a package from the distribution centre to a customer as the vector \(\vec{d} = 3\mathbf{i} + 5\mathbf{j} - 2\mathbf{k}\), where distances are measured in kilometres. a) Use scalar multiplication to determine the position vector of a checkpoint located \(\frac{2}{3}\) of the way along the direct path from the distribution centre to the customer. (1 mark) b) Determine the position vector if the drone needs to travel to a location that is 4 times as far in the opposite direction from the distribution centre. (1 mark) c) Use your result from part (b) to calculate the distance from the distribution centre to this new location, correct to two decimal places. (1 mark)

A vector is given by \(\mathbf{a} = 3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}\). Determine the vector \(-4\mathbf{a}\) in Cartesian form.

Use scalar multiplication to determine the vector \(3\vec{OB}\) where \(\vec{OB}\) is shown in the diagram below.