Question Video: Finding the Angle between Two Straight Lines in Two Dimensions Mathematics

Video Transcript

Find the measure of the acute angle included between the two straight lines 𝐿 sub one and 𝐿 sub two whose equations are 𝐫 equals two, negative seven plus 𝑘 times negative one, eight and 𝑥 equals three plus 12𝑑, 𝑦 equals four 𝑑 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 𝛼 is the measure of the acute angle between two straight lines of slopes 𝑚 sub one and 𝑚 sub two, then 𝛼 satisfies the equation the tan of 𝛼 is equal to the absolute value of 𝑚 sub one minus 𝑚 sub two over one plus 𝑚 sub one times 𝑚 sub two. Therefore, we can find the measure of the acute angle between the lines by finding the slopes of each line. Let’s 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 𝑦 sub one over 𝑥 sub one, provided 𝑥 sub one is nonzero. From the given vector equation, we see that 𝑥 sub one is negative one and 𝑦 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, 𝑥 equals 𝑎 sub zero plus 𝑏 sub zero 𝑑 and 𝑦 equals 𝑎 sub one plus 𝑏 sub one 𝑑, represents a line, then the quotient of the coefficients of the parameter 𝑑 tells us the slope of the line, provided 𝑏 sub zero is nonzero. We have that 𝑚 sub two will be equal to 𝑏 sub one divided by 𝑏 sub zero. We can now substitute the values from the given pair of parametric equations into this formula. We see that 𝑏 sub zero is 12 and 𝑏 sub one is four. So we find that 𝑚 sub two is equal to one-third.

We can now find an equation for 𝛼 by substituting these slopes into the formula for the measure of the acute angle between the two lines. We obtain that the tan of 𝛼 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 𝛼 is equal to five.

Since we want the measure of the acute angle between the lines, we can directly solve for 𝛼 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 𝛼 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 𝛼 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.