# Calculus 3 : Cross Product

## Example Questions

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### Example Question #1 : Cross Product

Let , and .

Find .

Explanation:

We are trying to find the cross product between  and .

Recall the formula for cross product.

If  , and , then

.

Now apply this to our situation.

### Example Question #2 : Cross Product

Let , and .

Find .

Explanation:

We are trying to find the cross product between  and .

Recall the formula for cross product.

If  , and , then

.

Now apply this to our situation.

### Example Question #3 : Cross Product

True or False: The cross product can only be taken of two 3-dimensional vectors.

False

True

True

Explanation:

This is true. The cross product is defined this way. The dot product however can be taken for two vectors of dimension n (provided that both vectors are the same dimension).

### Example Question #4 : Cross Product

Which of the following choices is true?

Explanation:

By definition, the order of the dot product of two vectors does not matter, as the final output is a scalar.  However, the cross product of two vectors will change signs depending on the order that they are crossed.  Therefore

.

### Example Question #5 : Cross Product

For what angle(s) is the cross product ?

Explanation:

We have the following equation that relates the cross product of two vectors  to the relative angle between them , written as

.

From this, we can see that the numerator, or cross product, will be  whenever .  This will be true for all even multiples of .  Therefore, we find that the cross product of two vectors will be  for .

### Example Question #6 : Cross Product

Evaluate

Explanation:

It is not possible to take the cross product of -component vectors. The definition of the cross product states that the two vectors must each have components. So the above problem is impossible.

### Example Question #7 : Cross Product

Compute .

Explanation:

To evaluate the cross product, we use the determinant formula

So we have

. (Use cofactor expansion along the top row. This is typically done when taking any cross products)

### Example Question #8 : Cross Product

Evaluate .

Explanation:

To evaluate the cross product, we use the determinant formula

So we have

. (Use cofactor expansion along the top row. This is typically done when taking any cross products)

### Example Question #9 : Cross Product

Find the cross product of the two vectors.

Explanation:

To find the cross product, we solve for the determinant of the matrix

The determinant equals

As the cross-product.

### Example Question #10 : Cross Product

Find the cross product of the two vectors.