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MATH / One line is described by

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problem

One line is described by \[\begin{pmatrix} -1 \\ -3 \\ -5 \end{pmatrix} + t \begin{pmatrix} 3 \\ k \\ 7 \end{pmatrix}.\]Another line is described by \[\begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix} + u \begin{pmatrix} 1 \\ 4 \\ 7 \end{pmatrix}.\]Find kk so that the lines are coplanar (i.e. there is a plane that contains both lines).
Plain-text mathematical notation (without MathML)
One line is described by
\[\begin{pmatrix} -1 \\ -3 \\ -5 \end{pmatrix} + t \begin{pmatrix} 3 \\ k \\ 7 \end{pmatrix}.\]Another line is described by
\[\begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix} + u \begin{pmatrix} 1 \\ 4 \\ 7 \end{pmatrix}.\]Find k so that the lines are coplanar (i.e. there is a plane that contains both lines).
Original LaTeX notation
One line is described by
\[\begin{pmatrix} -1 \\ -3 \\ -5 \end{pmatrix} + t \begin{pmatrix} 3 \\ k \\ 7 \end{pmatrix}.\]Another line is described by
\[\begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix} + u \begin{pmatrix} 1 \\ 4 \\ 7 \end{pmatrix}.\]Find $k$ so that the lines are coplanar (i.e. there is a plane that contains both lines).

level

Level 3

type

Precalculus

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Source and history

task_id
95197f16-4ace-5fcb-b262-a61532b2ed41
task_key
precalculus--test--95197f16-4ace-5fcb-b262-a61532b2ed41
task_revision_id
1
config / split
precalculus / test

Official source

initial import