Define and use unit vectors and the perpendicular unit vectors 𝒊̂, 𝒋̂ and 𝒌̂.

A vector \(\vec{v}\) has components \(\vec{v} = 6\hat{i} - 3\hat{j} + 2\hat{k}\). Express \(\vec{v}\) as a unit vector.

Consider the vector \(\vec{p} = 6\hat{i} - 3\hat{j} + 2\hat{k}\). a) Determine the magnitude of \(\vec{p}\). (1 mark) b) Express \(\vec{p}\) as a product of its magnitude and a unit vector. (1 mark) c) Verify that your unit vector from part b) has magnitude 1. (1 mark)

Three vectors are defined as follows: \(\vec{p} = 3\hat{i} - 4\hat{j} + 12\hat{k}\) \(\vec{q} = a\hat{i} + 2\hat{j} - 2\hat{k}\) \(\vec{r} = 6\hat{i} + b\hat{j} - 3\hat{k}\) where \(a, b \in \mathbb{R}\). The unit vector in the direction of \(\vec{p}\) is denoted \(\hat{p}\). Given that \(\hat{p} \cdot \vec{q} = 1\) and \(\hat{p} \cdot \vec{r} = 0\), determine the values of \(a\) and \(b\).