Calculate and use a unit vector, 𝒏̂, in three-dimensional space. 𝒏̂ = 𝒏 |𝒏|

Calculate the unit vector in the direction of \(\mathbf{p} + \mathbf{q}\).

Three position vectors \(\vec{OA}\), \(\vec{OB}\), and \(\vec{OC}\) are given in the table below. Calculate the unit vector in the direction of \(\vec{AB} + \vec{BC}\).

Given \(\mathbf{n} = 3\mathbf{i} - 4\mathbf{j} + 12\mathbf{k}\), calculate the unit vector \(\hat{\mathbf{n}}\) in the direction of \(\mathbf{n}\).

Consider the points \(P(3, -2, 1)\) and \(Q(-1, 4, 5)\) in three-dimensional space. (a) Determine the vector \(\overrightarrow{PQ}\). (1 mark) (b) Calculate the magnitude of \(\overrightarrow{PQ}\). (1 mark) (c) Find the unit vector in the direction of \(\overrightarrow{PQ}\). Express your answer in component form, correct to two decimal places. (2 marks)

A drone flies from point \(P\) to point \(Q\) in three-dimensional space. The position vector of \(P\) relative to the origin is \(\vec{OP} = 3\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}\) and the position vector of \(Q\) is \(\vec{OQ} = 7\mathbf{i} + 10\mathbf{j} + 4\mathbf{k}\). (a) Determine the displacement vector \(\overrightarrow{PQ}\). (1 mark) (b) Calculate the magnitude of \(\overrightarrow{PQ}\). (1 mark) (c) Hence, calculate a unit vector in the direction of \(\overrightarrow{PQ}\). (1 mark)