Show that the condition for the lines `x= a_1z+b_1,y = c_1z+d_1` and `x = a_2z+b_2, y = c_2z+ d_2` be perpendicular is `a_1a_2+c_1c_2=0`

If the lines x=a_(1)y+b_(1),z=c_(1)y+d_(1) and x=a_(2)y+b_(2),z=c_(2)y+d_(2) are perpendicular, prove that, 1+a_(1)a_(2)+c_(1)c_(2)=0 .

Show that the plane ax+by+ca+d=0 divides the line joining the points (x_1,y_1,z_1) & (x_2,y_2,z_2) in the ratio - (ax_1+by_1+cz_1+d_1)/(ax_2+by_2+cz_2+d_2)

Show that the plane ax + by + cz + d = 0 divides, the line segment joining the points (x_1, y_1, z_1) and (x_2, y_2, z_2) in the- ratio -(ax_1+by_1+cz_1+d)/(ax_2+by_2+cz_2+d) .

Perpendiculars are drawn from points on the line (x+2)/2=(y+1)/(-1)=z/3 to the plane x + y + z=3 The feet of perpendiculars lie on the line (a) x/5=(y-1)/8=(z-2)/(-13) (b) x/2=(y-1)/3=(z-2)/(-5) (c) x/4=(y-1)/3=(z-2)/(-7) (d) x/2=(y-1)/(-7)=(z-2)/5

Let the equation of the plane containing the line x-y - z -4=0=x+y+ 2z-4 and is parallel to the line of intersection of the planes 2x + 3y +z=1 and x + 3y + 2z = 2 be x + Ay + Bz + C=0 Compute the value of |A+B+C| .

If x ,y \ a n d \ z are not all zero and connected by the equations a_1x+b_1y+c_1z=0,a_2x+b_2y+c_2z=0 , and (p_1+lambdaq_1)x+(p_2+lambdaq_2)+(p_3+lambdaq_3)z=0 , show that lambda=-|[a_1,b_1,c_1],[a_2,b_2,c_2],[p_1,p_2,p_3]|-:|[a_1,b_1,c_1],[a_2,b_2,c_2],[q_1,q_2,q_3]|

The straight lines (x)/(a_(1))=(y)/(b_(1))=(z)/(c_(1)) and (x-2)/(a_(2))=(y-3)/(b_(2))=(z)/(c_(2)) will be parallel if -

Represent the following linear equations in matrix form: a_(1)x+b_(1)y+c_(1)z+d_(1)=0 , a_(2)x+b_(2)y+c_(2)z+d_(2)=0 and a_(3)x+b_(3)y_+c_(3)z+d_(3)=0

The point of intersection of the line passing through (0,0,1) and intersecting the lines x+2y+z=1,-x+y-2z=2 and x+y=2,x+z=2 with x y plane is a. (5/3,-1/3,0) b. (1,1,0) c. (2/3,1/3,0) d. (-5/3,1/3,0)

z_1a n dz_2 lie on a circle with center at the origin. The point of intersection z_3 of he tangents at z_1a n dz_2 is given by 1/2(z_1+( z )_2) b. (2z_1z_2)/(z_1+z_2) c. 1/2(1/(z_1)+1/(z_2)) d. (z_1+z_2)/(( z )_1( z )_2)