Question

Find the coordinates of a point \(P\) on the line and a vector \(\mathrm{v}\) parallel to the line. $$ \frac{x+3}{5}=\frac{y}{8}=\frac{z-3}{6} $$

Short answer

The coordinates of point P on the line are \((-3, 0, 3)\) and the vector \(\mathrm{v}\) parallel to the line is given by \( (5,8,6) \)

Step-by-step solution

Identify the coordinates of point P

Choose a parameter, say 't' and set it equal to the fractions in the equation. By setting t equal to any real number, you'll get the corresponding x, y and z coordinates. For instance, if \(t=0\), you'll get \(x=-3\), \(y=0\), and \(z=3\) . So, the coordinates for point P would be \((-3, 0, 3)\).

Identify the vector parallel to the line

In the equation \(\frac{x+3}{5}=\frac{y}{8}=\frac{z-3}{6}\), the coefficients of t in terms of x, y and z acts as the direction vectors of the line. Thus, the vector \(\mathrm{v}\) that is parallel to the line is \( (5,8,6) \)