Question

Find a set of parametric equations of the line. The line passes through the point (5,-3,-4) and is parallel to \(\mathbf{v}=\langle 2,-1,3\rangle\)

Short answer

The parametric equations of the line are \(x = 5 + 2t, y = -3 - t, z = -4 + 3t\).

Step-by-step solution

Identify the vector and point

The line is parallel to the vector \(v = \langle 2,-1,3 \rangle\) and passes through the point \(P = (5,-3,-4)\).

Formulate the equations

Given a vector and a point, the parametric equations for the line are given by \(x= x_0 + vt, y = y_0 + ut, z = z_0 + wt\), where \(x_0, y_0, z_0\) are the coordinates of the point and \(v,u,w\) are the components of the vector.

Substitute given values

Substitute the coordinates of the given point and the components of the vector into the formula. Thus you get the equations: \(x = 5 + 2t, y = -3 - t, z = -4 + 3t.\)