 Hello, and welcome to this screencast on section 9.4, the cross-product. This screencast is going to cover computing the cross-product. The cross-product of two vectors, u and v, in three-space, is denoted by the time symbol. The result of this cross-product is another vector given by the following expression. At first, this expression may look intimidating and difficult to remember, however, if we rewrite the expression using determinants of matrices, important structure emerges. We can think of a matrix as an arrangement of numbers and rows and columns. For example, this right here is a 2 by 2 matrix. The determinant of a 2 by 2 matrix is given by the following formula here. Note that we can think of this as the difference between the product of this diagonal, A times D, minus D times C. Using this, we can rewrite the cross-product formula using three determinants, and we can actually take this one step farther. This expression involving three determinants can be written as the single determinant of this 3 by 3 matrix. To get this matrix, we arrange the standard unit vectors i, j, and k across the top row, the components of u across the second row, and the components of v across the third row. The order which we arrange the matrix does matter. Also, the fact that u is the first vector written in the cross-product tells us that the components of u are placed in the second row, while the components of v are placed in the third row. Putting everything together, we can compute the cross-product using these three expressions. To get the determinant of this 3 by 3 matrix up here, we start in the top left corner with the standard unit vector i. If we cross out the entries in the row and the column containing i, we are left with a 2 by 2 matrix. If we look down here, this is the 2 by 2 matrix that appears on the next line that is multiplied with the standard unit vector i. We do something similar for j. If we cross out the column and row containing j, we're left with these entries and these entries, putting those together, gives us this 2 by 2 matrix that we multiply with j. Note that this term is always subtracted from the first term. To get the last term of this middle expression here, we do the same thing with k. If we cross out the column and the row containing k, we're left with a 2 by 2 matrix, which is this, which is multiplied by k on this next line. Then we add this term to the previous two terms. To get from this second line down to the last line, we use the determinant of a 2 by 2 matrix formula from the previous slide. Now that we know how to compute the cross product, let's look at some properties. First, the first property, we see that the cross product is not commutative. u times v is not equal to v times u, instead it is equal to negative v times u. This shows us that the order in which the vectors are written in the cross product matters. In the second property, we see that the cross product does distribute over vector addition. Next, we see that the cross product of two vectors that are parallel is equal to the zero vector. This is the vector with all zeros as components, which has no magnitude nor direction. Lastly, we see that the cross product is not associative. Again, this shows that the order in which the vectors are written in the cross product matters.