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problem

Let \[\mathbf{a} = \begin{pmatrix} 5 \\ -3 \\ -4 \end{pmatrix} \quad \text{and} \quad \mathbf{b} = \begin{pmatrix} -11 \\ 1 \\ 28 \end{pmatrix}.\]There exist vectors p\mathbf{p} and d\mathbf{d} such that the line containing a\mathbf{a} and b\mathbf{b} can be expressed in the form v=p+dt.\mathbf{v} = \mathbf{p} + \mathbf{d} t.Furthermore, for a certain choice of d\mathbf{d}, it is the case that for all points v\mathbf{v} lying on the same side of a\mathbf{a} that b\mathbf{b} lies on, the distance between v\mathbf{v} and a\mathbf{a} is tt. Find d\mathbf{d}.
Plain-text mathematical notation (without MathML)
Let
\[\mathbf{a} = \begin{pmatrix} 5 \\ -3 \\ -4 \end{pmatrix} \quad \text{and} \quad \mathbf{b} = \begin{pmatrix} -11 \\ 1 \\ 28 \end{pmatrix}.\]There exist vectors p and d such that the line containing a and b can be expressed in the form
v=p+dt.Furthermore, for a certain choice of d, it is the case that for all points v lying on the same side of a that b lies on, the distance between v and a is t.  Find d.
Original LaTeX notation
Let
\[\mathbf{a} = \begin{pmatrix} 5 \\ -3 \\ -4 \end{pmatrix} \quad \text{and} \quad \mathbf{b} = \begin{pmatrix} -11 \\ 1 \\ 28 \end{pmatrix}.\]There exist vectors $\mathbf{p}$ and $\mathbf{d}$ such that the line containing $\mathbf{a}$ and $\mathbf{b}$ can be expressed in the form
\[\mathbf{v} = \mathbf{p} + \mathbf{d} t.\]Furthermore, for a certain choice of $\mathbf{d}$, it is the case that for all points $\mathbf{v}$ lying on the same side of $\mathbf{a}$ that $\mathbf{b}$ lies on, the distance between $\mathbf{v}$ and $\mathbf{a}$ is $t$.  Find $\mathbf{d}$.

level

Level 5

type

Precalculus

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Source and history

task_id
9564f72c-6cce-5bdd-8d32-fbf5d7c8be77
task_key
precalculus--test--9564f72c-6cce-5bdd-8d32-fbf5d7c8be77
task_revision_id
1
config / split
precalculus / test

Official source

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