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))`

Prove that the st. line ax + by + c = 0 divides the join of (x_1, y_1) and (x_2, y_2) in the ratio -(ax_1 + by_1 +c)/(ax_2+ by_2+c) .

Find the ratio in which the plane ax+by+cz+d=0 divides the join of the points P(x_1,y_1,z_1) and Q(x_2,y_2,z_2) lying on the plane. Hence, show that the points P(1,-2,3) and Q(0,0,-1) lie on opposite sides of the plane 2x+5y+7z=3.

Find the scalar components and magnitude of the vector joining the points P(x_1, y_1, z_1) and Q(x_2, y_2, z_2)

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

Show that, if the axes are rectangular, the equations of the line through (x_1, y_1, z_1) at right angles to the lines: x/l_1=y/m_1=z/n_1,x/l_2=y/m_2=z/n_2 are frac{x-x_1}{m_1n_2-m_2n_1}=frac{y-y_1}{n_1l_2-n_2l_1}=frac{z-z_1}{l_1m_2-l_2m_1}

Find the reflection of the plane a\'x+b\'y+c\'z+d\'=0 in the plane ax+by+cz+d=0

The plane passing through the point (-2, -2, 2) and containing the line joining the points (1, 1, 1) and (1, -1, 2) makes intercepts of length a, b, c respectively the axes of x, y and z respectively, then

The line (x-x_1)/0=(y-y_1)/1=(z-z_1)/2 is

If a x_1^2+by_1^2+c z_1^2=a x_2^2+b y_2^2+c z_2^2=a x_3 ^2+b y_3 ^2+c z_3^ 2=d a x_2x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1= a x_1x_2+b y_1y_2+c z_1z_2=f, then prove that | x_1 y_1 z_1 x_2 y_2 z_2 x_3 y_3 z_3 |= (d-f){(d+2f)/(a b c)}^(1//2)

Find the ratio in which the line segment joining : (2,1,5) and (3,4,3) is divided by the plane x+y-z=1/2