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# Prove some vector identities using the “cab minus bac” formula

In the previous exercise (Section 13.14, Exercise #9) we proved the “cab minus bac” formula: Using this formula prove the following identities:

1. .
2. .
3. if and only if .
4. .

1. Proof. Using the cab minus bac formula with in place of , in place of , and in place of we have 2. Proof. Applying the cab minus back formula to each of the three terms in the sum we have So, putting these together we have 3. Proof. From cab minus bac we have Furthermore, since , we can apply bac minus cab to get Therefore, 4. Proof. From a previous exercise (Section 13.14, Exercise #7(d)) we know the identity . In this case we have in place of , in place of and in place of . This gives us # Prove the “cab minus bac” formula

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 an identity of cross products and the unit coordinate vectors

Prove that we have the identity: Proof. Let and compute, # Prove an identity for the angle between vectors in Cn

The angle between two vectors non-zero is defined by the equation The inequality we established in the previous exercise (Section 12.17, Exercise #6) show that there is a unique satisfying this equation. Prove that we have Proof. From the definition of we have But then we know from this exercise (Section 12.17, Exercise #3) that And, we know from this exercise (Section 12.17, Exercise #5) that Therefore, # Prove some facts about vectors in Cn

1. Prove that for we have 2. For non-zero vectors prove that 1. Proof. Let for . 2. Proof. Let for . Then, and Therefore, Therefore, # Prove yet another identity for vectors in Cn

Let be any two vectors. Prove that we have the following identity, Proof. Using our computations of and in the previous exercise (Section 12.17, Exercise #4) we have # Prove another identity for vectors in Cn

Let be any two vectors. Prove that we have the following identity: Proof. From the previous exercise (Section 12.17, #3) we have the identity This also give us Therefore, # Prove an identity for vectors in Cn

Let be any two vectors. Prove the following identity: Proof. We can compute, noting that by Theorem 12.11(a), # Use vector methods to prove the trig identity cos (a-b) = cos a cos b + sin a sin b

Prove the trig identity by taking the dot product of the vectors and .

Proof.
The angle between the vectors and is given by # Prove an identity of given finite sums

Prove the identity: Proof. Using the hint (that ) we start with the expression on the left, (The interchange of the sum and integral is fine since it is a finite sum. Those planning to take analysis should note that this cannot always be done in the case of infinite sums.) Now, we have a reasonable integral, but we still want to get everything back into the form of the sum on the right so we make the substitution , . This gives us new limits of integration from 1 to 0. Therefore, we have 