Question Video: Finding the Angle between Two Straight Lines in Two Dimensions Mathematics • First Year of Secondary School

Video Transcript

Find the measure of the acute angle between the following pair of straight lines: the vector negative nine, negative three plus 𝑠 times the vector negative one, negative six and the vector seven, negative seven added to 𝑠 times the vector four, negative 12.

In this question, we’re asked to determine the measure of the acute angle between two given straight lines, and we can note that both of these straight lines are given in vector form. To answer this question, let’s start by recalling the formula for determining the measure of the acute angle between any two given straight lines. We know if 𝛼 is the acute angle between two straight lines with slopes of 𝑚 sub one and 𝑚 sub two, then the tan of 𝛼 will be equal to the absolute value of 𝑚 sub one minus 𝑚 sub two divided by one plus 𝑚 sub one times 𝑚 sub two. And we can use this to determine the value of 𝛼. All we need to do is take the inverse tangent of both sides of the equation.

We can also notice something interesting about this formula. If the two lines are perpendicular, then one of two things will happen. Either 𝑚 sub one times 𝑚 sub two is negative one, in which case the denominator of the right-hand side of the equation is zero. So, the tan of 𝛼 will be undefined, which means that 𝛼 is 90 degrees. Or alternatively, one of the straight lines may be vertical. In this case, we can’t use this formula since the slope of this line would be undefined. So, we would need to use a different method.

Finally, we should also take care to make sure the two lines intersect since if the two lines are parallel, we may not want to define an angle between them. However, if the two lines are coincident, we do want to say that 𝛼 is zero. To apply this formula to answer this question, we need to determine the slope of the two lines. We can do this by recalling one result. If we have a line 𝐿 given in vector form, that’s the vector 𝐫 sub zero plus 𝑠 times the direction vector 𝐝, where 𝐝 is the vector 𝑎, 𝑏, then we know the slope of this line is 𝑏 divided by 𝑎, provided the value of 𝑎 is nonzero. If the value of 𝑎 is equal to zero, then we know our line is vertical. We can use this to determine the slope of both lines.

Let’s determine the slope of the first line. The direction vector of this line is the vector negative one, negative six. So, the slope of this line 𝑚 sub one will be negative six divided by negative one, which we can calculate is equal to six. We can calculate the slope of the second line in the same way, the direction vector of this line is the vector four, negative 12. So, the slope of this line 𝑚 sub two will be negative 12 divided by four, which we can calculate is negative three. And at this point, we can note that our two lines must intersect because they have different slopes.

Therefore, we can substitute the values of these two slopes into our equation and rearrange to determine the value of 𝛼. This gives us that the tan of 𝛼 is equal to the absolute value of six minus negative three divided by one plus six times negative three. Now, all we need to do is evaluate the right-hand side of this equation. First, in the numerator, six minus negative three is equal to six plus three, which is nine. Second, in the denominator, one plus six times negative three is one minus 18, which is negative 17. So, we get the absolute value of nine divided by negative 17. Finally, the absolute value of nine divided by negative 17 is just nine over 17. So, we’ve shown the tan of 𝛼 is nine over 17.

We can find the value of 𝛼 by taking the inverse tangent of both sides of the equation. Typing this into our calculator, where we make sure our calculator is set to degrees mode, gives us that 𝛼 is 27.897 and this expansion continues degrees. Now, we could stop here. However, we could also give our answer in terms of degrees, minutes, and seconds. To do this, we first need to recall there are 60 minutes in a degree and 60 seconds in a minute. We can then use this to convert our angle into degrees, minutes, and seconds.

First, we can see there are 27 degrees in this angle, so we’ll take out these 27 degrees and then consider the remaining expansion. That’s 0.897 and this expansion continues degrees. We want to determine how many minutes are in this remaining angle. To do this, we note there are 60 minutes in a degree. So if we multiply this angle by 60, we’ll convert it into minutes. Using the exact value, this gives us 53.836 and this expansion continues minutes. We can now see there are 53 minutes in this angle. So if we take out these 53 minutes, this leaves us with 0.836 and this expansion continues minutes. We want to determine what this is in seconds.

And we know to convert an angle from minutes into seconds, we need to multiply it by 60. Once again, making sure we use the exact value, we get 50.175 and this expansion continues seconds. We can then round our answer. We’ll round our answer to one decimal place. We know that the second decimal digit is seven. This means we need to round our value up. This then gives us an answer of 27 degrees, 53 minutes, and 50.2 seconds to the nearest one decimal place.