### Video Transcript

Find the measure of the acute angle
included between the two straight lines πΏ sub one and πΏ sub two whose equations
are π« equals two, negative seven plus π times negative one, eight and π₯ equals
three plus 12π, π¦ equals four π minus five, respectively, in terms of degrees,
minutes, and seconds, to the nearest second.

In this question, we are given the
equations of two straight lines, one in vector form and the other in parametric
form. We want to use these equations to
find the measure of the acute angle between the lines. We need to give our answer in
degrees, minutes, and seconds, to the nearest second.

To do this, we can recall that if
πΌ is the measure of the acute angle between two straight lines of slopes π sub one
and π sub two, then πΌ satisfies the equation the tan of πΌ is equal to the
absolute value of π sub one minus π sub two over one plus π sub one times π sub
two. Therefore, we can find the measure
of the acute angle between the lines by finding the slopes of each line. Letβs start with the line given in
vector form.

We can recall that the direction
vector of the line tells us the horizontal and vertical components of a vector
parallel to the line. In particular, the slope of the
line is given by π¦ sub one over π₯ sub one, provided π₯ sub one is nonzero. From the given vector equation, we
see that π₯ sub one is negative one and π¦ sub one is eight. So we can substitute these values
into the formula to see that the slope of the line is negative eight.

In the same way, we can recall that
if a pair of parametric equations, π₯ equals π sub zero plus π sub zero π and π¦
equals π sub one plus π sub one π, represents a line, then the quotient of the
coefficients of the parameter π tells us the slope of the line, provided π sub
zero is nonzero. We have that π sub two will be
equal to π sub one divided by π sub zero. We can now substitute the values
from the given pair of parametric equations into this formula. We see that π sub zero is 12 and
π sub one is four. So we find that π sub two is equal
to one-third.

We can now find an equation for πΌ
by substituting these slopes into the formula for the measure of the acute angle
between the two lines. We obtain that the tan of πΌ is
equal to the absolute value of negative eight minus one-third divided by one plus
negative eight times one-third. We can then evaluate the numerator
and denominator separately to obtain the absolute value of negative 25 over three
divided by negative five over three. We can evaluate this by canceling
the shared factor of negative five-thirds in the numerator and denominator to find
that the tan of πΌ is equal to five.

Since we want the measure of the
acute angle between the lines, we can directly solve for πΌ by taking the inverse
tangent of both sides of the equation. We do not need to worry about other
solutions to this equation. We have that πΌ is equal to the
inverse tan of five. We can find this value by using a
calculator set to degrees mode. This gives us that πΌ is equal to
78.69 and this expansion continues degrees.

We want to give our answer in
degrees, minutes, and seconds. And we can do this by pressing the
conversion button on our calculator. This gives us 78 degrees, 41
minutes, and 24.24 and this expansion continues seconds. We now need to round this to the
nearest second. This then gives us our final answer
that the measure of the acute angle between the two given straight lines to the
nearest second is 78 degrees, 41 minutes, and 24 seconds.