N2017 P1 Q6

Interpret geometrically the vector equation \(\mathbf{r}=\mathbf{a}+t\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are constant vectors and \(t\) is a parameter.[2]

Interpret geometrically the vector equation \(\mathbf{r}\cdot \mathbf{n}=d\), where \(\mathbf{n}\) is a constant unit vector and \(d\) is a constant scalar, stating what \(d\) represents.[3]

Given that \(\mathbf{b}\cdot \mathbf{n}\ne 0\), solve the equations \(\mathbf{r}=\mathbf{a}+t\mathbf{b}\) and \(\mathbf{r}\cdot \mathbf{n}=d\) to find \(\mathbf{r}\) in terms of \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{n}\) and \(d\). Interpret the solution geometrically.[3]