Write the equation of the plane containing the lines $\vec{r} = \vec{a} + λ \vec{b} and \vec{r} = \vec{a} + μ \vec{c} .$

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Solution

$The given plane passes through the lines \vec{r} = \vec{a} + λ \vec{b} and \vec{r} = \vec{a} + μ \vec{c} So, the plane passes through the vector \vec{a} and parallel to the vectors \vec{b} and \vec{c} . So, the plane passes through the vector \vec{a} whose normal vector is \vec{b} × \vec{a} . (It means that \vec{n} = \vec{b} × \vec{a}) So, the equation of plane in scalar product form is (\vec{r} - \vec{a}) . \vec{n} = 0 \Rightarrow (\vec{r} - \vec{a}) . (\vec{b} \times \vec{c}) = 0$

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Similar questions

\vec{r} = \vec{a} + λ \vec{b} + μ \vec{c}

r = a_{1} + λ b

r = a_{2} + μ b

r = a + λ (b \times c)

r = b + μ (c \times a)

\to r = \to a + λ (\to b \times \to c)

\to r = \to b + μ (\to c \times \to a)

(\to a, \to b, \to c

)

\vec{r} = \vec{a} + λ \vec{b}

\vec{r} . \vec{n} = 0 .

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