Understand and use vector notation: 𝐴𝐵⃗⃗⃗⃗⃗, 𝑐 ~, 𝒅 and unit vector notation 𝒏̂.

Consider the points A(2, −1, 4), B(5, 3, −2) and C(−1, 2, 7) in three-dimensional space. (a) Express the vector $\overrightarrow{AB}$ in component form. (b) Determine the magnitude of $\overrightarrow{AC}$. (c) Find the unit vector $\hat{\mathbf{n}}$ in the direction of $\overrightarrow{BC}$.

Consider points A(2, −1, 3) and B(5, 3, −2) in three-dimensional space. a) Express the vector $\overrightarrow{AB}$ in component form. (1 mark) b) Determine the magnitude of $\overrightarrow{AB}$. (1 mark) c) Express the unit vector $\hat{n}$ in the direction of $\overrightarrow{AB}$ in component form. Leave your answer in exact form. (1 mark)

Given the vector $\mathbf{v} = 3\mathbf{i} - 4\mathbf{j} + 12\mathbf{k}$, which of the following is the unit vector $\hat{\mathbf{v}}$ in the direction of $\mathbf{v}$?

In a three-dimensional navigation system, an aircraft departs from point P with position vector **p** = 3**i** + 2**j** + 5**k** (measured in kilometres from an origin O) and arrives at point Q with position vector **q** = 7**i** – 4**j** + 11**k**. a) Determine the displacement vector $\overrightarrow{PQ}$ in component form. (1 mark) b) Calculate the magnitude of $\overrightarrow{PQ}$ correct to two decimal places. (1 mark) c) Express the unit vector in the direction of $\overrightarrow{PQ}$ using the notation $\hat{\mathbf{n}}$. Leave your answer in exact form. (1 mark)

Points P, Q, R and S are shown in the diagram below. Determine which vector expression correctly represents $\overrightarrow{PS}$.