1. Determine vector and parametric equations of the line that passes through $A(2,-1,3)$ and is perpendicular to $\frac{x-3}{1}=\frac{y}{-2}, z=6$. (4 marks)
Answer
\(\boxed{\text{Vector Equation: } r(t) = (2, -1, 3) + t(1, 1/2, 0)}\)
Step 1 ：Find the direction vector of the given line using the coefficients of the parametric equations: \((1, -2, 0)\)
Step 2 ：Find a vector that is perpendicular to the direction vector by choosing values for a, b, and c such that their dot product is 0: \((1, 1/2, 0)\)
Step 3 ：Write the parametric equations of the line: \(x = 2 + t\), \(y = -1 + (1/2)t\), \(z = 3\)
Step 4 ：Write the vector equation of the line: \(r(t) = (2, -1, 3) + t(1, 1/2, 0)\)
Step 5 ：\(\boxed{\text{Parametric Equations: } x = 2 + t, y = -1 + (1/2)t, z = 3}\)
Step 6 ：\(\boxed{\text{Vector Equation: } r(t) = (2, -1, 3) + t(1, 1/2, 0)}\)
