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MATH / Let mathbf{a} = begin{pmatrix} 1 3 0 end{pmatrix}, mathbf{b} = begin{pmatrix} -11 5 2 end{pmatrix}, and mathbf{c} = begin{pmatrix} 1 + sqrt{5} 4 -5 end{pmatrix}. Find k if th…

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problem

Let $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 0 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} -11 \\ 5 \\ 2 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 1 + \sqrt{5} \\ 4 \\ -5 \end{pmatrix}.$ Find kk if the vectors a+b+c\mathbf{a} + \mathbf{b} + \mathbf{c} and 3(b×c)−8(c×a)+k(a×b)3 (\mathbf{b} \times \mathbf{c}) - 8 (\mathbf{c} \times \mathbf{a}) + k (\mathbf{a} \times \mathbf{b})are orthogonal.
Plain-text mathematical notation (without MathML)
Let $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 0 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} -11 \\ 5 \\ 2 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 1 + \sqrt{5} \\ 4 \\ -5 \end{pmatrix}.$  Find k if the vectors a+b+c and
3(b×c)−8(c×a)+k(a×b)are orthogonal.
Original LaTeX notation
Let $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 0 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} -11 \\ 5 \\ 2 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 1 + \sqrt{5} \\ 4 \\ -5 \end{pmatrix}.$  Find $k$ if the vectors $\mathbf{a} + \mathbf{b} + \mathbf{c}$ and
\[3 (\mathbf{b} \times \mathbf{c}) - 8 (\mathbf{c} \times \mathbf{a}) + k (\mathbf{a} \times \mathbf{b})\]are orthogonal.

level

Level 5

type

Precalculus

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Source and history

task_id
fa336db2-53d3-5fdd-b9b8-f9fd48081442
task_key
precalculus--test--fa336db2-53d3-5fdd-b9b8-f9fd48081442
task_revision_id
1
config / split
precalculus / test

Official source

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