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) = 2hat(i) + 2hat(j) + 3hat(k), vec(b) = -hat(i) + 2hat(j) + hat(k) and vec(c )=3hat(i)+2hat(j) such that vec(a)+lambda vec(b) is perpendicular to vec(c) , then find the value of lambda .

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) = 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 ) .

The scalar product of the vector vec(a) = hat(i) + hat(j) + hat(k) with a unit vector along the sum of vectors vec(b) = 2hat(i) + 4hat(j) - 5hat(k) and vec(c ) = lambda hat(i) + 2hat(j) + 3hat(k) is equal to one. Find the value of lambda and hence find the unit vector along vec(b) + vec(c ) .

If the vectors 2hat(i) + 3hat(j) - 6hat(k) and 4hat(i) - m hat(j) - 12 hat(k) are parallel find m.

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(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.

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) = hat(i) + hat(j) + hat(k), vec(b) = 4hat(i) - 2hat(j) + 3hat(k) and vec(c ) = hat(i) - 2hat(j) + hat(k) , find a vector of magnitude 6 units which is parallel to the vector 2vec(a) - vec(b) + 3vec(c ) .