Show that the straight lines whose direction cosines are given by the equations `a l+b m+c n=0, u(l)^2+v(m)^2+w (n)^2=0` are parallel or perpendicular as `(a^2)/u+(b^2)/v+(c^2)/w=0ora^2(v+w)+b^2(w+u)+c^2(u+v)=0.`

Find the angle between the lines whose direction cosines are given by the equation l + m + n = 0 and l^2 + m^2 - n^2 = 0.

Show that the straight lines whose direction cosines are given by 2l + 2m - n = 0 and mn + nl + Im = 0 are at right angles.

The angle between the lines whose direction cosines are given by l + m + n = 0 and l^2 = m^2 + n^2 is .....

Find the angle between the line whose direction cosines are given by l+m+n=0a n d2l^2+2m^2-n^2-0.

Find the direction cosines of the two lines which are connected by the relation l+m+n=0,mn−2nl−2lm=0

Show that two lines a_(1) x + b_(14) y + c_(1) = 0 and a_(2) x + b_(2) y + c_(2) = 0 , where b_(1) , b_(2) ne 0 are Perpendicular if a_(1) a_(2) + b_(1) b_(2) =0 .

The function u=e^x sinx ; v=e^x cos x satisfy the equation v(d u)/(dx)-u(d v)/(dx)=u^2+v^2 b. (d^2u)/(dx^2)=2v c. (d^2)/(dx^2)=-2u d. (d u)/(dx)+(d v)/(dx)=2v

Prove that the three lines from O with direction cosines l_1, m_1, n_1: l_2, m_2, n_2: l_3, m_3, n_3 are coplanar, if l_1(m_2n_3-n_2m_3)+m_1(n_2l_3-l_2n_3)+n_1(l_2m_3-l_3m_2)=0

A U-shaped conducting frame is placed in a magnetic field B in such a way that the plane of the frame is perpendicular to the field lines. A conducting rod is supported on the parallel arms of the frame, perpendicular to them and is given a velocity v_0 at time t = 0. Prove that the velocity of the rod at time t will be given by v_t=v_0 "exp" ((-B^2l^2)/(mR)t) .