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.

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)

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

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)

If the vectors vec (a) = 2 hat (i) - hat (j) + hat (k) , vec ( b) = hat (i) + 2 hat (j) - 3 hat (k) and vec(c ) = 3 hat (i) + lambda hat (j) + 5 hat (k) are coplanar , find the value of lambda

Let vec(a) = hat (i) + hat (j) + hat (k) , vec (b) = hat (i) and vec (c) = c_(1)hat (i) + c_(2)hat (j)+ c_(3)hat(k) then , if c_(1)=1 and c_(2) = 2 , find c_(3) which makes vec(a),vec(b) and vec(c ) coplanar .

In each of the following show that the given vectors are coplanar: vec(a) = hat (i) + hat (j) - 6 hat (k) , vec(b) = hat (i) + 3 hat (j) + 4 hat (k) , vec(c) = 2 hat (i) + 5 hat (j) + 3 hat (k)

If the vectors ahat (i) + a hat (j) + c hat (k) , hat (i) + hat (k) and c hat (i) + c hat (j) + b hat (k) be coplanar, show that c^(2) = ab

If the vectors vec(a) = 2 hat (i) - lambda hat(j) + 3 hat (k), vec(b) = 3 hat (i) + 2 hat (k) - mu hat (k) and vec(c ) = hat (i) + hat (j) + hat (k) are coplanar , find mu in terms of lambda

Find the angle between the vectors vec a= hat i+hat j-hat k and vec b=hat i-hat j+hat k