Let \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) and \(\vec{d}\) be the vectors. Consider the following: I. \((\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})\) II. \((\vec{a}\times\vec{b})\times(\vec{c}\times\vec{d})\) III. \((\vec{a}\cdot\vec{b})\cdot(\vec{c}\cdot\vec{d})\) IV. \((\vec{a}\cdot\vec{b})\times(\vec{c}\cdot\vec{d})\) V. \(\{(\vec{a}\times\vec{b})\cdot\vec{c}\}\times\vec{d}\) where '\(\cdot\)' represents scalar product of vectors and '\(\times\)' represents vector product of vectors. How many of the above are not well-defined?
Three
Expressions I and II combine the vectors \(\vec{a}\times\vec{b}\) and \(\vec{c}\times\vec{d}\) using the dot product and cross product respectively, so both are well-defined. In III and IV, \(\vec{a}\cdot\vec{b}\) and \(\vec{c}\cdot\vec{d}\) are scalars, and neither the dot product nor the cross product is defined between two scalars, so both are not well-defined. In V, \((\vec{a}\times\vec{b})\cdot\vec{c}\) is a scalar, and the cross product of a scalar with the vector \(\vec{d}\) is not defined. So III, IV and V — three expressions — are not well-defined.
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Select the correct answer using the code given below.
Important Questions from Scalar Triple Product
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is 15 then α = ?
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