The “cab minus bac” formula is the vector identity
Let and . Prove that
This is the “cab minus bac” formula in the case . Prove similar formulas for the special cases and . Put these three results together to prove the formula in general.
Proof. For the case we have
Similarly, for and we have
So, if is any vector in then we have
Prove that we have the identity:
Proof. Let and compute,
Prove that we have the formula
Proof. Let and . Then we compute,
Compute the volume of the parallelpiped determined by the vectors .
The volume is given by the scalar triple product
Find all such that the vectors
are linearly dependent.
The vectors are linearly dependent if and only if the scalar triple product
So, we compute
So, the vectors are dependent for all such that . Thus the vectors are dependent for
Consider unit length, orthogonal vectors , and a vector such that
Prove the following.
- and are orthogonal and the length of is .
- The vectors form a basis for .
- Proof. We compute,
since and since and are orthogonal by assumption. Thus, and are orthogonal. Next,
since by hypothesis and . Hence, from the vector equation we have
- Proof. Since and are orthogonal (part (a)), we know the vectors are independent. Thus, they form a basis for since any three independent vectors in are a basis
- Proof. We compute, the vector is given by
Then the three coordinates of this cross product are given by
Expanding these out we obtain the coordinates
Since we know and since we know . So, simplifying the expressions above, for each of the coordinates we have
Hence, we indeed have
- Proof. We compute