Video Transcript
Calculate the cross product of 𝐲
equals two 𝐢 plus four 𝐣 plus eight 𝐤 and 𝐳 equals six 𝐢 plus four 𝐣 plus two
𝐤.
In this example, we want to solve
for 𝐲 cross 𝐳, where 𝐲 and 𝐳 are both three-dimensional vectors. We can begin our solution by
recalling the mathematical definition for a cross product. The cross product of two vectors,
we can call them 𝐀 and 𝐁, is equal to the determinant of a three-by-three matrix ,
where the columns are headed by the three unit vectors 𝐢, 𝐣, and 𝐤. And the last two rows are populated
by the respective components of vectors 𝐀 and 𝐁. We can apply this formula to our
vectors 𝐲 and 𝐳. As we set up our matrix, we know
the top row will be the 𝐢, 𝐣, and 𝐤 unit vectors. The next row will be the respective
components of the vector 𝐲. Looking at 𝐲, we see that it’s
𝑥-component is two, its 𝑦-component is four, and its 𝑧-component is eight. So we write those values in to our
matrix. In the next row, we’ll write the
components of 𝐳. The 𝑥-component of 𝐳 is six. Its 𝑦-component is four. And its 𝑧-component is two.
Now we’re ready to calculate the
determinant of this matrix and solve for the cross product of 𝐲 cross 𝐳. When we compute this determinant,
we find that the 𝐢-component is eight minus 32, or negative 24. The 𝐣-component is four minus 48,
or negative 44. And the 𝐤-component is eight minus
24, which equals negative 16. Our overall cross product then is
negative 24𝐢 minus 44𝐣 minus 16𝐤. That’s the cross product of the
vectors 𝐲 and 𝐳.
