a. Show that the vector v = ai + bj is perpendicular to the line ax + by = c by establishing that the slope of v is the negative reciprocal

Question

a. Show that the vector v = ai + bj is perpendicular to the line ax + by = c by establishing that the slope of v is the negative reciprocal of the slope of the given line.
b. Determine the slope of the vector : v = ai + bj

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Reagan 1 week 2022-01-11T14:59:01+00:00 1 Answer 0 views 0

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    2022-01-11T15:00:53+00:00

    Answer:

    Part A:

    m_1m_2=-1

    \frac{b}{a}(\frac{-a}{b})=-1\\-1=-1

    Hence proved that Vector= ai + bj is perpendicular to the line ax + by = c.

    Part B:

    Slope of vector = \frac{b}{a}

    Step-by-step explanation:

    Condition for perpendicular is:

    m_1m_2=-1

    Part A:

    Consider the vector v = ai + bj

    x component of vector=a

    y component of vector=b

    Slope of vector=m_1=\frac{y}{x}=\frac{b}{a}

    Consider the line ax + by = c:

    Rearranging the equation:

    ax+by=c

    by=c-ax

    y=\frac{-ax}{b}+\frac{c}{b}

    According to general equation of line: y=mx+c

    Where m is the slope

    In our case the slope of above line is:

    m_2=\frac{-a}{b}

    According to the condition of perpendicular:

    m_1m_2=-1

    \frac{b}{a}(\frac{-a}{b})=-1\\-1=-1

    Hence proved that Vector= ai + bj is perpendicular to the line ax + by = c.

    Part B:

    Slope of vector is also calculated above.

    Since the slope of vector is negative reciprocal of the slope of the given line:

    According to equation of line ax + by = c

    y=\frac{-ax}{b}+\frac{c}{b}

    According to  general equation of line: y=mx+c

    Where m is the slope

    Slope of given line=m=\frac{-a}{b}

    negative reciprocal of the slope of the given line = \frac{b}{a}

    Slope of vector = \frac{b}{a}

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