The equation of the plane passing through the point (1, 2, -3) and perpendicular to the planes

3x + y - 2z = 5 and 2x - 5y - z = 7, is :

Given, equation of planes are <br/><br/>3x + y - 2z = 5 <br/><br/>2x - 5y - z = 7 <br/><br/>and point ( 1, 2, 3). <br/><br/>Normal vector of required plane = n = $$\left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 3 & 1 & { - 2} \cr 2 & { - 5} & { - 1} \cr } } \right|$$ <br/><br/>= ${\widehat i}$(-1 - 10) - ${\widehat j}$( -3 + 4) + ${\widehat k}$( -15 - 2) <br/><br/>= -11${\widehat i}$ - ${\widehat j}$ - 17${\widehat k}$ <br/><br/>Now, the equation of plane passing through (1, 2, -3) having normal vector -11${\widehat i}$ - ${\widehat j}$ - 17${\widehat k}$ is <br/><br/>-[11(x - 1) + (y - 2) + 17(z + 3)] = 0 <br/><br/>$\Rightarrow$ 11x + y + 17z + 38 = 0