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 

There are a few approaches to answering this.

One is to notice that the right hand side of each equation is some number times some number times some other number, whereas is a vector. It is not possible for a vector to equal a number (or a "scalar" technically), so

, and

are out of the question.

is not correct since the right hand side of the equation is the definition of the dot product of two vectors, which is not represented by the left hand side.

is true. While it's not the definition of the cross product, it is a formula used to find the area of the parallelogram formed by vectors  .

To evaluate the cross product, we use the determinant formula

.

So we have

A quick way to answer this is to note that the two vectors point in the same direction (one is a scalar multiple of the other). Hence they are parallel, and their cross product is the zero vector.

We can still use the formula

to evaluate it as well.

We have

The cross product is the determinant of the following matrix:

Which is equal to:

Which, in bracket form, is the same as

The cross product of the two vectors

is defined as the determinant of the matrix

For the vectors in the problem we solve the determinant of the matrix

which is

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:

We must first 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:

Writing this in vector notation, we get

To start, 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:

which simplified becomes