Show that the plane `ax+by+cz+d=0` divides the line joining `(x_1, y_1, z_1) and (x_2, y_2, z_2)` in the ratio of `(-(ax_1+ay_1+cz_1+d)/(ax_2+by_2+cz_2+d))`

In which ratio the plane x + y + z = 5 divides line segment joining points (2, -1, 3) and (-1, 2, 1) ?

If point P divided line segment joining points A(x_1,y_1,z_1) and B(x_2,y_2,z_2) in ratio k : 1 then co - ordinates on P.

If the intercept of a line between the coordinate axes is divided by the point (-5, 4) in the ratio 1:2 , then find the equation of the line. Thinking Process : Coordinates of the point which divides line segment joining (x_1 , y_1 ) and ( x_2, y_2) in ratio (m_1 : m_2) ((m_1 x_2 + m_2 x_1)/( m_1 + m_2) , ( m_1 y_2 + m_2 y_1)/( m_1 + m_2)) .

Find the ratio in which 2x+3y+5z=1 divides the line joining the points (1, 0, -3) and (1, -5, 7).

The length of projection of the line segmet joining the points (1, 0, -1) and (-1, 2, 2) on the plane x+3y-5z=6 is equal to

If the distance of the plane x - y + z + lambda = 0 from the point (1, 1, 1) is d_1 and the distance of this point from the origin is d_2 and d_2d_2 = 5 then find the value of lambda .

Let a_1x+b_1y+c_1z+d_1=0 and a_2x+b_2y+c_2z+d_2=0 be two planes, where d_1, d_2gt0 . Then, origin lies in acute angle, If a_1a_2+b_1b_2+c_1c_2lt0 and origin lies in obtuse angle if a_1a_2+b_1b_2+c_1c_2gt0 . Further point (x_1, y_1, z_1) and origin both lie either in acute angle or in obtuse angle. If ( a_1x_1+b_1y_1+c_1z_1+d_1)(a_2x_1+b_2y_1+c_2z_1+d_2)gt0 . one of (x_1, y_1, z_1) and origin in lie in acute and the other in obtuse angle,If ( a_1x_1+b_1y_1+c_1z_1+d_1)(a_2x_1+b_2y_1+c_2z_1+d_2)lt0 Q. Given the planes x+2y-3z+2=0 and x-2y+3z+7=0 . If a point P(1, 2, 2), then

Let a_1x+b_1y+c_1z+d_1=0 and a_2x+b_2y+c_2z+d_2=0 be two planes, where d_1, d_2gt0 . Then, origin lies in acute angle, If a_1a_2+b_1b_2+c_1c_2lt0 and origin lies in obtuse angle if a_1a_2+b_1b_2+c_1c_2gt0 . Further point (x_1, y_1, z_1) and origin both lie either in acute angle or in obtuse angle. If ( a_1x_1+b_1y_1+c_1z_1+d_1)(a_2x_1+b_2y_1+c_2z_1+d_2)gt0 . one of (x_1, y_1, z_1) and origin in lie in acute and the other in obtuse angle,If ( a_1x_1+b_1y_1+c_1z_1+d_1)(a_2x_1+b_2y_1+c_2z_1+d_2)lt0 Q. Given the planes x+2y-3z+5=0 and 2x+y+3z+1=0 . If a point P(2, -1, 2). Then

Let a_1x+b_1y+c_1z+d_1=0 and a_2x+b_2y+c_2z+d_2=0 be two planes, where d_1, d_2gt0 . Then, origin lies in acute angle, If a_1a_2+b_1b_2+c_1c_2lt0 and origin lies in obtuse angle if a_1a_2+b_1b_2+c_1c_2gt0 . Further point (x_1, y_1, z_1) and origin both lie either in acute angle or in obtuse angle. If ( a_1x_1+b_1y_1+c_1z_1+d_1)(a_2x_1+b_2y_1+c_2z_1+d_2)gt0 . one of (x_1, y_1, z_1) and origin in lie in acute and the other in obtuse angle,If ( a_1x_1+b_1y_1+c_1z_1+d_1)(a_2x_1+b_2y_1+c_2z_1+d_2)lt0 Q. Given that planes 2x+3y-4z+7=0 and x-2y+3z-5=0 . If a point P(1, -2, 3), then