I'm a little confused on this problem. The Question asks to find the cross product, $a \times b$.
$$a = \langle 6,0,-2\rangle\text{ and }b = \langle 0,8,0\rangle$$
The formula that is given in my text book is $a \times b = \langle a_2 b_3 – a_3 b_2, a_3 b_1 – a_1 b_3, a_1 b_2 – a_2 b_1\rangle$
And I get $\langle 16, -8, 48\rangle$
However, the book says that the answer is $16i + 48k$? How are they getting that? Could someone tell me what I doing wrong, and also what exactly the cross product is supposed to be. I understand that is is a vector product, but can someone explain that in more details?
Thanks in advance.
Best Answer
If you look under Matrix Product in Wikipedia, the cross product is $$\left |\begin {array} {c c c} i&j&k\\6&0&-2\\0&8&0 \end {array}\right |$$ The zeros eliminate a lot of terms, leaving $-i(-2\cdot 8)+k(6\cdot 8)$.
Added: to take a $3 \times 3$ matrix determinant you multiply the elements in each of the diagonals down and to the right (including the broken ones) and subtract the products of the diagonals up and to the right. So this is $i\cdot 0 \cdot 0 + j \cdot (-2) \cdot 0 + k \cdot 6 \cdot 8 -i \cdot (-2)\cdot 8 - j\cdot 6 \cdot 0 - k \cdot 0 \cdot 0$ The particular terms are a bit harder to follow because of the several zeros, but it should be clear.