How do you find a vector equation when given a scalar equation?

For example: 3x-5z+15=0, how would i find the vector equation of this?|||Did you mean to write "z" instead of "y"? If so we are dealing in three dimensions and the equation is that of a plane, not a line. I will assume what you wrote is correct.





The normal vector n, of the plane can be taken from the coefficients of the variables.





n = %26lt;3, 0, -5%26gt;





Now we need to find a point in the plane. Let z = 0 and solve for x. y can be anything so let y = 0 also.





3x + 15 = 0


3x = -15


x = -5





So we have the point P(-5, 0, 0).





Define an arbitrary point in the plane R(x,y,z). Then the vector PR lies in the plane. The normal vector is orthogonal to any vector that lies in the plane. And the dot product of orthogonal vectors is zero.





n 鈥?PR = 0


n 鈥?%26lt;R - P%26gt; = 0


%26lt;3, 0, -5%26gt; 鈥?%26lt;x + 5, y - 0, z - 0%26gt; = 0