Video Transcript
Find the measure of the angle between the two straight lines 𝐿 one: 𝑥 is equal to five minus eight 𝑡, 𝑦 is equal to negative three minus four 𝑡, 𝑧 is equal to five plus six 𝑡 and 𝐿 two: 𝑥 minus five over three is equal to 𝑦 plus five over negative six, which is equal to 𝑧 minus two over negative two, and round it to the nearest second.
To find the angle between two lines in space, since the angle between them is the angle between their direction vectors, we first find their direction vectors. We can then use the given formula to find the angle between the two lines. cos 𝜃 is equal to the magnitude of the dot product of 𝐝 one and 𝐝 two divided by the magnitude of 𝐝 one multiplied by the magnitude of 𝐝 two.
In this question, line 𝐿 one is given in parametric form, where 𝑥 is equal to 𝑥 one plus 𝜆𝑎, 𝑦 is equal to 𝑦 one plus 𝜆𝑏, and 𝑧 is equal to 𝑧 one plus 𝜆𝑐. 𝐿 two, on the other hand, is given in Cartesian form, where 𝑥 minus 𝑥 one over 𝑎 is equal to 𝑦 minus 𝑦 one over 𝑏, which is equal to 𝑧 minus 𝑧 one over 𝑐, where 𝜆 is a scalar. And the line passes through the point 𝐴 with coordinates 𝑥 one, 𝑦 one, 𝑧 one, with direction vector 𝐝 equal to 𝑎𝐢 plus 𝑏𝐣 plus 𝑐𝐤.
In line 𝐿 one, our values of 𝑎, 𝑏, and 𝑐 are negative eight, negative four, and six, respectively. So 𝐝 one is equal to negative eight 𝐢 minus four 𝐣 plus six 𝐤, which can be written as negative eight, negative four, six as shown. Considering the Cartesian form of line 𝐿 two, we see that 𝑎 is equal to three, 𝑏 is equal to negative six, and 𝑐 is equal to negative two. So the direction vector of this line 𝐝 two is equal to three, negative six, negative two.
We are now in a position to find the dot product and magnitude of these vectors. To find the dot or scalar product of two vectors, we find the sum of the products of the corresponding components. In this case, we have negative eight multiplied by three plus negative four multiplied by negative six plus six multiplied by negative two. This simplifies to negative 24 plus 24 plus negative 12, which is equal to negative 12. Next, we find the magnitude of 𝐝 one and 𝐝 two. This is equal to the square root of the sum of the squares of the individual components. So the magnitude of the direction vector 𝐝 one is equal to the square root of 64 plus 16 plus 36, which is the square root of 116. Since 116 is equal to four multiplied by 29, using our laws of radicals, we can rewrite this as two root 29.
In the same way, the magnitude of 𝐝 two is equal to the square root of three squared plus negative six squared plus negative two squared. This is equal to the square root of nine plus 36 plus four, which is the square root of 49 and in turn equals seven. Noting that we must take the magnitude of the dot product, which is equal to positive 12, we can substitute our values back into our formula for cos 𝜃. This is equal to 12 divided by two root 29 multiplied by seven. The right-hand side simplifies to 12 over 14 root 29, and we can then take the inverse cosine of both sides. This is equal to 80.841425 and so on degrees.
We are asked to give our answer to the nearest second. One way of doing this is using the degrees, minutes, and seconds button on our calculator, giving us an answer of 80 degrees, 50 minutes, and 29 seconds to the nearest second. Alternatively, to find the minutes and seconds of the angle, we successively multiply the decimal part by 60. Using either method, we find that the measure of the angle between the two lines 𝐿 one and 𝐿 two is 80 degrees, 50 minutes, and 29 seconds.
