Video Transcript
If vector 𝐀 is equal to four,
negative two, negative nine and vector 𝐁 is equal to four, three, four, determine
the cross product of vector 𝐀 and vector 𝐁.
We begin by recalling that when we
calculate the cross product of two vectors in three dimensions, the answer is a
vector perpendicular to the original two vectors. The cross product is equal to the
determinant of the following three-by-three matrix. In the top row, we have the unit
vectors 𝐢, 𝐣, and 𝐤. In the middle row, we have the
components of the vector 𝐀, and in the bottom row we have the individual components
of the vector 𝐁.
This determinant is calculated in
three steps. Firstly, we multiply the unit
vector 𝐢 by 𝐴 sub 𝑦 𝐵 sub 𝑧 minus 𝐴 sub 𝑧 𝐵 sub 𝑦. Inside the parentheses, we have the
determinant of the two-by-two matrix created when we cross out the first row and
first column of the three-by-three matrix. Our second term is negative 𝐣
multiplied by 𝐴 sub 𝑥 𝐵 sub 𝑧 minus 𝐴 sub 𝑧 𝐵 sub 𝑥. Finally, we have the unit vector 𝐤
multiplied by 𝐴 sub 𝑥 𝐵 sub 𝑦 minus 𝐴 sub 𝑦 𝐵 sub 𝑥.
In this question, the vectors 𝐀
and 𝐁 are four, negative two, negative nine and four, three, four,
respectively. 𝐀 cross 𝐁 is therefore equal to
the determinant of the three-by-three matrix shown. This is equal to the unit vector 𝐢
multiplied by negative eight minus negative 27. We then subtract the unit vector 𝐣
multiplied by 16 minus negative 36. Finally, we add the unit vector 𝐤
multiplied by 12 minus negative eight. Negative eight minus negative 27 is
equal to 19. 16 minus negative 36 is equal to
52, so the second term is negative 52𝐣. Finally, 12 minus negative eight is
equal to 20.
If vector 𝐀 is equal to four,
negative two, negative nine and vector 𝐁 is equal to four, three, four, then 𝐀
cross 𝐁 is equal to 19𝐢 minus 52𝐣 plus 20𝐤. This could also be written in
component form as 19, negative 52, 20.
