91Ó°ÊÓ

In three-dimensional space, a cross product is a binary operation on two vectors. It yields a vector orthogonal to both vectors.$\mathbf{a}\mathbf{×}\mathbf{b}$indicates the vector product of two vectors, aand b.

The given vector equation is $\mathrm{V}=\mathrm{aA}+\mathrm{bB}.$

Find a and b .

Find the cross product of A with V.

$\begin{array}{ll}\mathrm{A}×\mathrm{V}=\mathrm{A}×\left(\mathrm{aA}+\mathrm{bB}\right)& \\ \mathrm{A}×\mathrm{V}=\mathrm{bA}×8& \left(âˆ\mu \mathrm{A}×\mathrm{A}=0\right)\end{array}$

Find the cross product of B with V.

$\begin{array}{ll}\mathrm{B}×\mathrm{v}=\mathrm{B}×\left(\mathrm{aA}+\mathrm{bB}\right)& \\ 8×\mathrm{V}=\mathrm{aB}×\mathrm{A}& \left(âˆ\mu \mathrm{B}×8=0\right)\end{array}$

Find the dot product of$(A×\mathrm{V})$ and n.

$$\begin{gathered} (\mathrm{A}×\mathrm{V})â‹\dots \mathrm{n}=\mathrm{b}\left(\right(\mathrm{A}×\mathrm{B})â‹\dots \mathrm{n}) \\ \mathrm{b}=\frac{(\mathrm{A}×\mathrm{V})â‹\dots \mathrm{n}}{(\mathrm{A}×\mathrm{B})â‹\dots \mathrm{n}} \end{gathered}$$

Find the dot product of$(B×V)$ and n .

$$\begin{gathered} (\mathrm{B}×\mathrm{V})â‹\dots \mathrm{n}=\mathrm{a}\left(\right(\mathrm{B}×\mathrm{A})â‹\dots \mathrm{n}) \\ \mathrm{a}=\frac{(\mathrm{B}×\mathrm{V})â‹\dots \mathrm{n}}{(\mathrm{B}×\mathrm{A})â‹\dots \mathrm{n}} \end{gathered}$$

Therefore, any vector V in a plane can be written as a linear combination of two non-parallel vectors A and B in the plane; such that $\mathrm{V}=\mathrm{aA}+\mathrm{bB}$if $\mathrm{a}=\frac{\left(\mathrm{B}×\mathrm{v}\right)·\mathrm{n}}{\left(\mathrm{B}×\mathrm{A}\right)·\mathrm{n}}$and role="math" localid="1659073525714" $\mathrm{b}=\frac{\left(\mathrm{A}×\mathrm{v}\right)·\mathrm{n}}{\left(\mathrm{A}×\mathrm{B}\right)·\mathrm{n}}$,