Home » dot products

# 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 