If r-x-y-zk- find r-r-xy

Given: r⃗=xî+yĵ+zk̂ (r⃗×î).(r⃗×ĵ)+xy =[(xî+yĵ+zk̂)×î].[(xî+yĵ+zk̂)×ĵ]+xy =[(xî×î+yĵ×î+zk̂×î)].[(xî×ĵ+yĵ×ĵ+zk̂×ĵ)]+xy =(0 y+zĵ).(xk̂+0+0) ...

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Composite Function

If r=xî+yĵ+...

Question

If $r = x^i + y^j + z^k$ find $(\to r \times^i) (\to r \times^j) + x y$

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Solution

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\to r = x^i + y^j + z^k

(\to r \times^i) . (\to r \times j) + x y .

^i \times ((^a -^j) \times^i) +^j \times ((^a -^k) \times^j) +^k \times ((^a -^i) \times^k) = \to 0

\to a = x^i + y^j + z^k,

8 (x^{3} - x y + z x)

\to r = x^i + y^j + z^k

\to a = x^i + y^j + z^k

(\to a \times^i) . (^i +^j) + (\to a \times^j) . (^j +^k) + (\to a \times^k) . (^k \times^i) =

{\to a}_{1} = x^i -^j +^k

{\to a}_{2} =^i + y^j + 2^k

x^i + y^j + z^k

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