hexaedru8p

2022-10-09

If a and c are unit vectors at an angle $\pi /3$ with each other and $\left(a×\left(b×c\right)\right).\left(a×c\right)=5$, then what is the value of [a b c]? I just know the basic meaning of what are vector and scalar triple product. How do I do this question?

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Salma Baird

Expert

We want to compute $a\cdot b×c=-b\cdot a×c$. We'll use $a×\left(b×c\right)=\left(a\cdot c\right)b-\left(a\cdot b\right)c$ so
$5=\left(a×\left(b×c\right)\right)\cdot \left(a×c\right)=\left(a\cdot c\right)\left(b\cdot a×c\right).$
Hence
$\left[abc\right]=\frac{-5}{a\cdot c}=\frac{-5}{\mathrm{cos}\frac{\pi }{3}}=-10.$

Still Have Questions?

Shawn Peck

Expert

Use $a×\left(b×c\right)=\left(a\cdot c\right)b-\left(a\cdot b\right)c$. Then
$\left(a×\left(b×c\right)\right)\cdot \left(a×c\right)=\left(a\cdot c\right)b\cdot \left(a×c\right)-\left(a\cdot b\right)c\cdot \left(a×c\right)=-\left(a\cdot b\right)\left[a,b,c\right]$
etc.

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