3.32. A useful rule of thumb for the cubic system is that a given [hk] direction is the normal to the (hkl) plane. Using this rule and Equation 3.3, determine which members of the (110) family of directions lie within the (111) plane. (Hint: The dot product of two per- pendicular vectors is zero.) 3.33. Which members of the (111) family of directions lie within the (110) plane? (See the comments in Problem 3.32.) 3. EXAMPLE 3.11 What is the angle between the [110] and [111] directions in the cubic system? SOLUTION From Equation 3.3. 8 arccos uu' + vv' + ww' Vu + v + wV(u') + (v) + (w') 1+1+0 23 = arccos arccos 0.816 35.3. htl