Prove that vectors ` vec u=(a l+a_1l_1) hat i+(a m+a_1m_1) hat j+(a n+a_1n_1) hat k` ` vec v=(b l+b_1l_1) hat i+(b m+b_1m_1) hat j+(b n+b_1n_1) hat k` ` vec w=(b l+b_1l_1) hat i+(b m+b_1m_1) hat j+(b n+b_1n_1) hat k` are coplanar.

Prove that vectors vec u=(a l+a_1l_1) hat i+(a m+a_1m_1) hat j+(a n+a_1n_1) hat k vec v=(b l+b_1l_1) hat i+(b m+b_1m_1) hat j+(b n+b_1n_1) hat k vec w=(c l+c_1l_1) hat i+(c m+c_1m_1) hat j+(c n+c_1n_1) hat k are coplanar.

Prove that vectors vec u=(a l+a_1l_1) hat i+(a m+a_1m_1) hat j+(a n+a_1n_1) hat k vec v=(b l+b_1l_1) hat i+(b m+b_1m_1) hat j+(b n+b_1n_1) hat k vec w=(c l+c_1l_1) hat i+(c m+c_1m_1) hat j+(c n+c_1n_1) hat k are coplanar.

Prove that vectors vec u=(a l+a_1l_1) hat i+(a m+a_1m_1) hat j+(a n+a_1n_1) hat k vec v=(b l+b_1l_1) hat i+(b m+b_1m_1) hat j+(b n+b_1n_1) hat k vec w=(c l+c_1l_1) hat i+(c m+c_1m_1) hat j+(c n+c_1n_1) hat k are coplanar.

Prove that vectors vec u=(al+a_(1)l_(1))hat i+(am+a_(1)m_(1))hat j+(an+a_(1)n_(1))hat kvec v=(bl+b_(1)l_(1))hat i+(bm+b_(1)m_(1))hat j+(bn+b_(1)n_(1))hat kvec w=(bl+b_(1)l_(1))hat i+(bm+b_(1)m_(1))hat j+(bn+b_(1)n_(1))hat k are coplanar.

If hat (i),hat (j) and hat (k) are unit vectors along three mutuaaly perpendicular axes and vec (a) = a_(1) hat (i) + a_(2) hat(j) + a_(3) hat (k) , vec(b) = b_(1) hat (i) + b_(2) hat (j) + b_(3) hat (k) and vec ( c ) = c_(1) hat(i) + c_(2) hat (j) + c_(3) hat (k) prove that vec(a) . (vec(b)+vec(c)) = vec(a).vec(b) + vec(a).vec(c)

If hat (i),hat (j) and hat (k) are unit vectors along three mutuaaly perpendicular axes and vec (a) = a_(1) hat (i) + a_(2) hat(j) + a_(3) hat (k) , vec(b) = b_(1) hat (i) + b_(2) hat (j) + b_(3) hat (k) and vec ( c ) = c_(1) hat(i) + c_(2) hat (j) + c_(3) hat (k) prove that (vec(b)+vec(c))xx vec(a) = vec(b) xx vec(a) + vec(c) xx vec(a)

If vec (a) = a_(1) hat (i) + a_(2) hat (j) + a_(3) hat (k) , vec(b) = b_(1) hat (i) + b _(2) hat (j) + b_(3) hat (k) and vec (c ) = c_(1) hat (i) + c_(2) hat (j) + c_(3) hat (k) prove that vec (a ) xx ( vec (b) + vec (c ) ) = vec ( a) xx vec ( b) + vec (a) xx vec(c)

Let V be the volume of the parallelopiped formed by the vectors vec a = a_1 hat i +a_2 hat j +a_3 hat k and vec b =b_1 hat i +b_2 hat j +b_3 hat k and vec c = c_1 hat i + c_2 hat j + c_3 hat k . If a_r, b_r and c_r , where r = 1, 2, 3, are non-negative real numbers and sum_(r=1)^3(a_r+b_r+c_r)=3L show that V le L^3

Let V be the volume of the parallelepiped formed by the vectors vec a = a_i hat i +a_2 hat j +a_3 hat k and vec b =b_1 hat i +b_2 hat j +b_3 hat k and vec c = c_1 hat i + c_2 hat j + c_3 hat k . If a_r, b_r and c_ r, where r = 1, 2, 3, are non-negative real numbers and sum_(r=1)^3(a_r+b_r+c_r)=3L show that V le L^3

Two balls , having linear momenta vec(p)_(1) = p hat(i) and vec(p)_(2) = - p hat(i) , undergo a collision in free space. There is no external force acting on the balls. Let vec(p)_(1) and vec(p)_(2) , be their final momenta. The following option(s) is (are) NOT ALLOWED for any non -zero value of p , a_(1) , a_(2) , b_(1) , b_(2) , c_(1) and c_(2) (i) vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) + c_(1) hat(k) , vec(p)_(2) = a_(2) hat(i) + b_(2) hat(j) (ii) vec(p)_(1) = c_(1) vec(k) , vec(p)_(2) = c_(2) hat(k) (iii) vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) + c_(1) hat(k) ,vec(p)_(2) = a_(2) hat(i) + b_(2) hat(j) - c_(1) hat(k) (iv) vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) , vec(p)_(2) = a_(2) hat(i) + b_(1) hat(j)