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

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Stumbling Robot

A Fraction of a Dot
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Tag: dot products

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

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Prove some properties of the scalar triple product

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Compute the angle between *(1,0,i,i,i)* and *(i,i,i,0,i)* in *C*^{5}

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Compute some dot products of vectors in *C*^{2}

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

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Compute some vector algebra expressions

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

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

- .
- .
- .
- .

*Proof.*We havesince for any vectors (in this case and

*Proof.*Using part (a), we have*Proof.*We have*Proof.*We have

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

Compute the angle between and in .

We compute as follows:

Let

be vectors in . Compute the following dot products.

- ;
- ;
- ;
- ;
- ;
- ;
- ;
- ;
- ;
- .

- We compute,
- We compute,
- We compute,
- We compute,
- We compute,
- We compute,
- We compute,
- We compute,
- We compute,
- We compute,

Prove or disprove the following statement about vectors in :

If and , then .

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

but, .

Let

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

- ;
- ;
- ;
- ;
- .

- First, we have means , then we compute
- First, we have means , then we compute
- First, we have means , then we compute
- First, we have means , then we compute
- First, we have means , then we compute