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MATH / Consider two lines: line l parameterized as

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problem

Consider two lines: line ll parameterized as \begin{align*} x &= 1 + 4t,\\ y &= 4 + 3t \end{align*}and the line mm parameterized as \begin{align*} x &=-5 + 4s\\ y &= 6 + 3s. \end{align*}Let AA be a point on line ll, BB be a point on line mm, and let PP be the foot of the perpendicular from AA to line mm. Then BP→\overrightarrow{BP} is the projection of BA→\overrightarrow{BA} onto some vector $\begin{pmatrix} v_1\\v_2\end{pmatrix}$ such that v1+v2=−7v_1+v_2 = -7. Find $\begin{pmatrix}v_1 \\ v_2 \end{pmatrix}$.
Plain-text mathematical notation (without MathML)
Consider two lines: line l parameterized as
\begin{align*} 
x &= 1 + 4t,\\
y &=  4 + 3t
\end{align*}and the line m parameterized as
\begin{align*} 
x &=-5  + 4s\\
y &=  6 + 3s.
\end{align*}Let A be a point on line l, B be a point on line m, and let P be the foot of the perpendicular from A to line m.

Then (BP)→ is the projection of (BA)→ onto some vector $\begin{pmatrix} v_1\\v_2\end{pmatrix}$ such that v₁+v₂=−7. Find $\begin{pmatrix}v_1 \\ v_2 \end{pmatrix}$.
Original LaTeX notation
Consider two lines: line $l$ parameterized as
\begin{align*} 
x &= 1 + 4t,\\
y &=  4 + 3t
\end{align*}and the line $m$ parameterized as
\begin{align*} 
x &=-5  + 4s\\
y &=  6 + 3s.
\end{align*}Let $A$ be a point on line $l$, $B$ be a point on line $m$, and let $P$ be the foot of the perpendicular from $A$ to line $m$.

Then $\overrightarrow{BP}$ is the projection of $\overrightarrow{BA}$ onto some vector $\begin{pmatrix} v_1\\v_2\end{pmatrix}$ such that $v_1+v_2 = -7$. Find $\begin{pmatrix}v_1 \\ v_2 \end{pmatrix}$.

level

Level 5

type

Precalculus

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Source and history

task_id
20ab45e6-3ace-54ad-b03e-e41b6b46c8dc
task_key
precalculus--test--20ab45e6-3ace-54ad-b03e-e41b6b46c8dc
task_revision_id
1
config / split
precalculus / test

Official source

initial import