Question

Find the coordinates of a point \(P\) on the line and a vector \(\mathrm{v}\) parallel to the line. $$ x=4 t, \quad y=5-t, \quad z=4+3 t $$

Short answer

The coordinates of point \(P\) on the line are \(P(0,5,4)\), and the coordinates of the vector \(v\) parallel to the line are \(v(4, -1, 3)\).

Step-by-step solution

Find the point P on the line

Choose a value for \(t\), this will provide a specific point on the line. It can be any real number, but for simplicity, it's often easiest to choose \(t = 0\). This will yield \(P(0,5,4)\).

Find the vector v parallel to the line

The coefficients of \(t\) in the parametric equations are the coordinates for the direction vector of the line. Therefore, the vector \(v\) would be \((4,-1,3)\).

Format the solution

The coordinates of point \(P\) are \(P(0,5,4)\) and the coordinates of vector \(v\) parallel to the line are \(v(4,-1,3)\).