The vectors \(\vec a\,\,and\,\,\vec b\) are not perpendicular and \(\vec c\,\,and\,\,\vec d\) are two vector satisfying : \(\vec b \times \vec c = \vec b \times \vec d\,\,and\,\,\vec a\,\,\cdot\,\,\vec d = 0\) . Then the vector \(\vec d\) is equal to :

Engineering

Mathematics

Vector Product of Two Vectors

Question

The vectors \(\vec a\,\,and\,\,\vec b\) are not perpendicular and \(\vec c\,\,and\,\,\vec d\) are two vector satisfying :

\(\vec b \times \vec c = \vec b \times \vec d\,\,and\,\,\vec a\,\,\cdot\,\,\vec d = 0\) . Then the vector \(\vec d\) is equal to :

\(\vec c - \left( {\frac{{\vec a\,\,\cdot\,\,\vec c}}{{\vec a\,\,\cdot\,\,\vec b}}} \right)\,\vec b\)

\(\vec b - \left( {\frac{{\vec b\,\,\cdot\,\,\vec c}}{{\vec a\,\,\cdot\,\,\vec b}}} \right)\,\vec c\)

\(\vec c + \left( {\frac{{\vec a\,\,\cdot\,\,\vec c}}{{\vec a\,\,\cdot\,\,\vec b}}} \right)\,\vec b\)

\(\vec b + \left( {\frac{{\vec b\,\,\cdot\,\,\vec c}}{{\vec a\,\,\cdot\,\,\vec b}}} \right)\,\vec c\)

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\(\vec a \cdot \vec b \ne 0,\vec b \times \vec c = \vec b \times \vec d,\vec a \cdot \vec d = 0\)

\((\vec b \times \vec c) \times \vec a = (\vec b \times \vec d) \times \vec a\)

\((\vec b \cdot \vec a)\vec c - (\vec c \cdot \vec a)\vec b = (\vec b \cdot \vec a)\vec d - (\vec d \cdot \vec a)\vec b\)

\(\vec d = \vec c - \left( {\frac{{\vec a \cdot \vec c}}{{\vec a \cdot \vec b}}} \right)\vec b\)

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