Question

3. If four vectors a, b, c, and d all lie in the same plane, show that: (ax b) x (cx d) = 0. 4. Prove that (For simplicity work in Cartesian systern): (a) (AT)r = A (b) rdr = 0

Answer 1

a, b, c, d all lie in one plane implies RTP/(a times b) times (c times d) = 0 we know that a times b = P (say) and p will be perpendicular to a and b i.e. p perpendicular a p perpendicular b Similarly c times d = Q (say) Q perpendicular c q perpendicular d This would mean that P and Q will be either parallel or anti parallel as a, b, c. d all lie on the same plane implies P times Q = PQ Sin 0 downarrow 0 or PQ sin 180 downarrow 0 therefore (a times b) times (c times d) = P times Q = 0 \r\n (a) (A nabla) r = 0 Proof A = Ax i + Ay j + Az k nabla = i partial differential/partial differential x + j partial differential/partial differential y + k partial differential/partial differential z r = xi + yj + zk implies (A nabla) r = (Ax partial differential/partial differential x + Ay partial differential/partial differential y