Find the unit vector perpendicular to both the vectors `vec(a) + vec(b)` and `vec(a) - vec(b)` where `vec(a) = hat(i) + hat(j) + hat(k)` and `vec(b) = hat(i) + 2hat(j) + 3hat(k)`.

Find the unit vector perpendicular to each of the vectors vec(a)+vec(b)" and "vec(a)-vec(b)" where "vec(a)=3hat(i)+2hat(j)+2hat(k)" and "vec(b)=hat(i)+2hat(j)-2hat(k) .

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

Write a unit vector in the direction of the sum of vectors vec(a) = 2hat(i) - hat(j) + 2hat(k) and vec(b) = -hat(i) + hat(j) + 3hat(k) .

Write a unit vector in the direction of the sum of vectors vec(a) = 2hat(i) + 2hat(j) - 5hat(k) and vec(b) = 2hat(i) + hat(j) - 7hat(k) .

Write the projection of vec(b) + vec(c ) on vec(a) , where vec(a) = 2hat(i) - 2hat(j) + hat(k), vec(b) = hat(i) + 2hat(j) - 2hat(k) and vec(c ) = 2hat(i) - hat(j) + 4hat(k) .

Obtain the projection of the vector vec(a) = 2hat(i) + 3hat(j) + 2hat(k) on the vector vec(b) = hat(i) + 2hat(j) + hat(k) .

Find the unit vector perpendicular to the plane ABC where the position vectors A, B and C are 2hat(i) - hat(j) + hat(k), hat(i) + hat(j) + 2hat(k) and 2hat(i) + 3hat(k) .

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

Find the vector vec(p) which is perpendicular to both vec(alpha) = 4hat(i) + 5hat(j) - hat(k) and vec(beta) = hat(i) - 4hat(j) + 5hat(k) and vec(p).vec(q) = 21 , where vec(q) = 3hat(i) + hat(j) - hat(k) .

Find 'lambda' when the projection of vec(a) = lambda hat(i) + hat(j) + 4hat(k) on vec(b) = 2hat(i) + 6hat(j) + 3hat(k) is 4 units.