Question

Find sets of (a) parametric equations and (b) symmetric equations of the line through the point parallel to the given vector or line. (For each line, write the direction numbers as integers.) $$ (0,0,0) \quad \mathbf{v}=\left\langle-2, \frac{5}{2}, 1\right\rangle $$

Short answer

The parametric equations for the line are \(x = -2t\), \(y = \frac{5}{2}t\), and \(z = t\). The symmetric equations for the line are \(x/-2 = y/2.5 = z\).

Step-by-step solution

Formulate the Parametric Equations

The parametric equations of a line that begins at the point (0,0,0) and is parallel to the vector \(\mathbf{v}\) are given by \(x = x_0 + at\), \(y = y_0 + bt\), and \(z = z_0 + ct\), where \(a, b, c\) are the components of vector \(\mathbf{v}\), (x_0,y_0,z_0) is the point and \(t\) is a variable parameter. Here, the point is (0,0,0) and vector \(\mathbf{v}=-2\mathbf{i}+\frac{5}{2}\mathbf{j}+ \mathbf{k}\). Then, the parametric equations are \(x = 0 -2t\), \(y = 0 + \frac{5}{2}t\), and \(z = 0 + t\) or equivalent to \(x = -2t\), \(y = \frac{5}{2}t\), and \(z = t\).

Formulate the Symmetric Equations

Symmetric equations are obtained by isolating the variable parameter \(t\) in each of the parametric equations and then setting them equal to each other: \(t = -x/2 = y/(5/2) = z/1 = y/2.5\). So, the symmetric equations are \(x/-2 = y/2.5 = z\).