The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

With these principles in mind, we can calculate the cross product of our vectors and

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

With these principles in mind, we can calculate the cross product of our vectors and

To find the cross product of two vectors, we must write the determinant of the vectors:

Next, we take the cross product. One can do this by multiplying across from the top left to the lower right, and continuing downward, and then subtracting the terms multiplied from top right to the bottom left:

The vector is written in unit vector notation. We simply take the coefficients of our unit vectors and correspond them to x, y, and z:

To find the cross product of two vectors and , you find the determinant of the 3x3 matrix

Using this formula, we evaluate using the vectors from the problem statement:

First, we must write the determinant in order to take the cross product of the two vectors:

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

Next, we take the cross product. One can do this by multiplying across from the top left to the lower right, and continuing downward, and then subtracting the terms multiplied from top right to the bottom left:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

With these principles in mind, we can calculate the cross product of our vectors and

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

.

To find the cross product of two vectors and , you find the determinant of the 3x3 matrix

Using this formula, we evaluate using the vectors from the problem statement:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

With these principles in mind, we can calculate the cross product of our vectors and

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.