Answer
The first part of the question uses BAC-CAB rule to prove the three triple vector product sum up to zero.
For second part of the question, the condition is B ^ (A ^ C) = 0 where A,B,C are vectors.
Work Step by Step
For the first part, you express the triple products in dot product and scalar multiplication form, i.e. using BAC-CAB rule. (If you are unsure about this rule, it is a very important rule shown in the textbook).
For the second part, there are two ways of finding the condition. You can either observe from the first part where the three triple product sum up to zero, and if you want two of them to be negative of each other, then you want the third triple product to be zero. Another way is to use BAC-CAB rule again to prove as shown in the image.
