Given two vectors `A = -4hat(i) + 4hat(j) + 2hat(k)` and `B = 2hat(i) - hat(j) - hat(k)`. The angle made by (A+B) with `hat(i) + 2hat(j) - 4hat(k)` is :

Given two vectors vec(A) = -hat(i) + 2hat(j) - 3hat(k) and vec(B) = 4hat(i) - 2hat(j) + 6hat(k) . The angle made by (A+B) with x-axis is :

Three vectors vec(A) = 2hat(i) - hat(j) + hat(k), vec(B) = hat(i) - 3hat(j) - 5hat(k) , and vec(C ) = 3hat(i) - 4hat(j) - 4hat(k) are sides of an :

If angle bisector of vec(a) = 2hat(i) + 3hat(j) + 4hat(k) and vec(b) = 4hat(i) - 2hat(j) + 3 hat(k) is vec(c) = alpha hat(i) + 2hat(j) + beta hat(k) then :

Find the shortest distance between the lines : vec(r) = (4hat(i) - hat(j)) + lambda(hat(i) + 2hat(j) - 3hat(k)) and vec(r) = (hat(i) - hat(j) + 2hat(k)) + mu (2hat(i) + 4hat(j) - 5hat(k))

The vector(s) which is/are coplanar with vectors hat(i)+hat(j)+2hat(k) and hat(i)+2hat(j)+hat(k) are perpendicular to the vector hat(i)+hat(j)+hat(k) is are

The sine of the angle between 3hat(i)+hat(j)+2hat(k)and 2hat(i)-2hat(j)+4hat(k) is

vec(r )=(-4hat(i)+4hat(j) +hat(k)) + lambda (hat(i) +hat(j) -hat(k)) vec(r)=(-3hat(i) -8hat(j) -3hat(k)) + mu (2hat(i) +3hat(j) +3hat(k))

The position vectors of three points A,B anc C (-4hat(i) +2hat(j) -3hat(k)) (hat(i) +3hat(j) -2hat(k)) "and " (-9hat(i) +hat(j) -4hat(k)) respectively . Show that the points A,B and C are collinear.

A unit vector perpendicular to each of the vectors 2hat(i) - hat(j) + hat(k ) and 3hat(i) - 4hat(i) - hat(k) is