Video Transcript
Find to the nearest second the measure of the angle between a straight line with direction ratio five, three, two and a line with direction angles 47 degrees; 111 degrees; and 50 degrees, 23 minutes, and 30 seconds.
In this question, we’re given the direction ratio of one line, which we will call 𝐿 sub one, and the direction angles of a second line, which we will call 𝐿 sub two. In order to find a measure of the angle between the two lines, we will use the formula cos 𝜃 is equal to the absolute value of 𝑙 one, 𝑙 two plus 𝑚 one, 𝑚 two plus 𝑛 one, 𝑛 two, where 𝑙 one, 𝑚 one, 𝑛 one and 𝑙 two, 𝑚 two, 𝑛 two are the direction cosines of the lines 𝐿 sub one and 𝐿 sub two, respectively.
Let’s begin by considering how we can calculate these direction cosines from the information given. Beginning with 𝐿 sub one, we have the direction ratio five, three, two. And we know that the direction cosine for the 𝑥-component is given by 𝑙 one is equal to cos 𝛼, which is equal to 𝑎 over the square root of 𝑎 squared plus 𝑏 squared plus 𝑐 squared, where 𝛼 is the angle the direction vector of the line makes with the 𝑥-axis.
Substituting in our values of 𝑎, 𝑏, and 𝑐, we have 𝑙 one is equal to five over the square root of five squared plus three squared plus two squared. This simplifies to five over the square root of 38. And rationalizing the denominator by multiplying the numerator and denominator by root 38, we have 𝑙 one is equal to five root 38 over 38.
We can then repeat this process for our 𝑦- and 𝑧-components. The 𝑦-component 𝑚 one is equal to cos 𝛽, which is equal to 𝑏 over the square root of 𝑎 squared plus 𝑏 squared plus 𝑐 squared. This simplifies to three root 38 over 38. Finally, the 𝑧-component 𝑛 one is equal to two root 38 over 38. This means that for line 𝐿 sub one, the direction cosines are five root 38 over 38, three root 38 over 38, and two root 38 over 38.
Next, we consider line 𝐿 sub two, where we are given the direction angles. 𝛼 is equal to 47 degrees. This means that the 𝑥-component 𝑙 two is equal to the cos of 47 degrees. Likewise, since 𝛽 is equal to 111 degrees, 𝑚 two is equal to the cos of 111 degrees. The third direction angle 𝛾 is given in degrees, minutes, and seconds. Since there are 60 minutes in a degree and 3600 seconds in a degree, we can rewrite this as 50 plus 23 over 60 plus 30 over 3600 degrees, which approximates, to five decimal places, to 50.39167 degrees.
𝑛 two, the third direction cosine, for line 𝐿 sub two is therefore equal to the cosine of this. And we are now in a position to find the cosine of the angle between our two lines. Substituting in the values of the direction cosines, we have the following expression for cos of 𝜃. And typing the right-hand side of our equation into our calculator, we have 0.5856128 and so on.
Our next step is to take the inverse cosine of both sides of the equation, giving us 𝜃 is equal to 54.153704 and so on degrees. We are asked to give the measure of the angle to the nearest second. We can do this directly on the calculator, or by successively multiplying the decimal parts by 60. Either way, to the nearest second, the measure of the angle between the two lines is 54 degrees, nine minutes, and 13 seconds.
