Question

Find the coordinates of a point \(P\) on the line and a vector \(\mathrm{v}\) parallel to the line. $$ \frac{x-7}{4}=\frac{y+6}{2}=z+2 $$

Short answer

The coordinates of point P on the line are (7, -6, -2) and the vector v parallel to the line is <4, 2, 1>.

Step-by-step solution

Identify the equations of the line and isolate variables

First, notice that the equation given is in the form of an equivalent ratios, which means that each fraction must be equal. Isolate each variable to get \(x = 4t + 7\), \(y = 2t - 6\), and \(z = t - 2\). Now we need to find the coordinates of any point on the line, as well as the vector parallel to the line.

Calculate the coordinates of point P

To find point P on the line, we can set \(t\) to be any real value. For simplicity, let's choose \(t = 0\). Substituting \(t = 0\) into the equations above, we have \(x = 7\), \(y = -6\), and \(z = -2\). So, point P(7, -6, -2) is on the line.

Determine the vector parallel to the line

From our initial isolated equations, we can notice that the coefficients 4, 2, and 1 in the equations are actually the direction ratios of the line which gives the direction of vector v parallel to the line. Therefore, \(\mathrm{v} = <4, 2, 1>\).