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# Prove an identity relating scalar triple products of vectors A,B,C

Prove the following identity holds for vectors .

Proof. From a previous exercise (Section 13.14, Exercise #9(d)), we know

So, with in place of and in place of in this formula we have

# Prove some properties of the scalar triple product

Use the properties of the cross product and the dot product to prove the following properties of the scalar triple product.

1. .
2. .
3. .
4. .

1. Proof. We have

since for any vectors (in this case and

2. Proof. Using part (a), we have

3. Proof. We have

4. Proof. We have

# Compute the angle between (1,0,i,i,i) and (i,i,i,0,i) in C5

We define the angle between two vectors and in by the formula

Compute the angle between and in .

We compute as follows:

# Compute some dot products of vectors in C2

Let

be vectors in . Compute the following dot products.

1. ;
2. ;
3. ;
4. ;
5. ;
6. ;
7. ;
8. ;
9. ;
10. .

1. We compute,

2. We compute,

3. We compute,

4. We compute,

5. We compute,

6. We compute,

7. We compute,

8. We compute,

9. We compute,

10. We compute,

# Show that the cancellation law does not hold for the dot product

Prove or disprove the following statement about vectors in :

If and , then .

This statement is false. Let , and . Then we have

but, .

# Compute some vector algebra expressions

Let

Compute each of the following (and insert appropriate parentheses to make sensible expressions):

1. ;
2. ;
3. ;
4. ;
5. .

1. First, we have means , then we compute

2. First, we have means , then we compute

3. First, we have means , then we compute

4. First, we have means , then we compute

5. First, we have means , then we compute