(a) $[2,-3,5], [4,-5,1]$
Solution:
$[2,-3,5] \times [4,-5,1] = [22,18,2]$
No idea how they got $ [22,18,2]$
$\endgroup$ 12 Answers
$\begingroup$Cross product of two vectors is a vector which is orthogonal to both vectors. In $\mathbf{w} = \mathbf{u} \times \mathbf{v}$, $\mathbf{w}$ is orthogonal to both $\mathbf{u}$ and $\mathbf{v}$.
And to calculate cross product you may use the formula. For vectors $\mathbf{u}=[u_1, u_2, u_3 ]$ and $\mathbf{v}=[v_1, v_2, v_3]$: $$ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} i & j & k \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix} $$
and then expand the determinant from Row 1 ($i$, $j$ and $k$ are unit vectors).
$\endgroup$ $\begingroup$Suppose the vector is $(x, y, z)$. Orthogonality means \begin{align*} 2x - 3y + 5z = 0 \\ 4x - 5y + z = 0 \end{align*} Now fix $x$ (or $y$ or $z$) and solve for $y$ and $z$ in terms of $x$. This gives $22 y = 18 x$ and $22z = 2x$. Taking any nonzero $x$ yields the solution. Of course, $x=22$ would result in integer entries.
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