If `vec(a) = hat(i) - hat(j) + 7hat(k)` and `vec(b) = 5hat(i) - hat(j) + lambda hat(k)` , then find the value of `lambda` so that `vec(a) + vec(b)` and `vec(a) - vec(b)` are perpendicular vectors.

If vec(AB) = 3hat(i) + 2hat(j) + 6hat(k), vec(OA) = hat(i) - hat(j) - 3hat(k) , find the value of vec(OB) .

If vec(a) = x hat(i) + 2hat(j) - zhat(k) and vec(b) = 3hat(i) - y hat(j) + hat(k) are two equal vectors, then write the value of x+y+z.

If vec(a) = hat(i) + hat(j) + hat(k) and vec(b) = hat(j) - hat(k) , find a vector vec(c ) , such that vec(a) xx vec(c ) = vec(b) and vec(a).vec(c ) = 3 .

If vec(a)=2hat(i)+lambda hat(j)+hat(k) and vec(b)=-hat(i)+2hat(j)-3hat(k) are orthogonal, then value of lambda is

If vec(a) = 2hat(i) - hat(j) + 3hat(k) and vec(b) = (6hat(i) + lambda hat(j) + 9 hat(k)) and vec(a) is parallel to vec(b) , find the value of lambda .

If vec(a) = 3hat(i) - hat(j) and vec(b) = 2hat(i) + hat(j) - 3hat(k) then express vec(b) in the form of vec(b) = vec(b_(1)) + vec(b_(2)) where vec(b_(1))||vec(a) and vec(b_(2)) perpendicular to vec(a) .

If a hat(i) + 6hat(j) - hat(k) and 7hat(i) -3hat(j) + 17hat(k) are perpendicular vectors, then the value of a= ….

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

If vec(a) = 2hat(i) - 3hat(j) + hat(k), vec(b) = -hat(i) + hat(k), vec(c ) = 2hat(j) - hat(k) are three vectors, find the area of the parallelogram having diagonals vec(a) + vec(b) and vec(b) + vec(c ) .