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 )`.

Let vec(a) = hat(i) + 4hat(j) + 2hat(k), vec(b) = 3hat(i) - 2hat(j) + 7hat(k) and vec(c )= 2hat(i) - hat(j) + 4hat(k) . Find a vector vec(p) which is perpendicular to both vec(a) and vec(b) and vec(p).vec(c ) = 18 .

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) + 4hat(k), vec(b) = hat(i) + 2hat(j) - 3hat(k) and vec(c ) = 3hat(i) + 4hat(j) - hat(k) , then find vec(a).(vec(b) xx vec(c )) and (vec(a) xx vec(b)).vec(c ) . Is, vec(a).(vec(b) xx vec(c )) = (vec(a) xx vec(b)).vec(c ) ?

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) - 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 .

Find a vector of magnitue 8 units in the direction of the vector, vec(a) = 5hat(i) - hat(j) + 2hat(k)

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(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 vector bar(AB) = 2 hat(i) - hat(j) + hat(k) and bar(OB) = 3 hat(i) - 4hat(j) + 4 hat(k) , find the position vector bar(OA)

Let vec(a) =hat(i) +2hat(j) +hat(k),vec(b) =hat(i) -hat(j) +hat(k) and vec(c ) =hat(i) +hat(j) -hat(k) , vector in the plane of vec(a) and vec(b) whose projection on vec(c ) is 1/(sqrt(3)) is ______.