Find the measure of the angle between the straight line 𝑥 equals one, 𝑦 equals two and the straight line 𝑦 equals negative one, 𝑧 is equal to zero.
In this question, we’re given two straight lines: the straight line 𝑥 is equal to one, 𝑦 is equal to two and the straight line 𝑦 is equal to negative one and 𝑧 is equal to zero. And we need to determine the measure of the angle between these two straight lines.
So let’s start by recalling how we determine the angle between two straight lines in space. We recall if 𝜃 is the angle between two straight lines 𝐿 sub one and 𝐿 sub two with direction vectors 𝐝 sub one and 𝐝 sub two, then the cos of 𝜃 will be equal to the dot product of 𝐝 sub one and 𝐝 sub two divided by the magnitude of 𝐝 sub one times the magnitude of 𝐝 sub two. In other words, we can just find the angle between two straight lines by finding the angle between their direction vectors.
So to answer this question, we’re going to need to find the direction vectors of the two given straight lines. Let’s start by finding the direction vector of the first line. There’s a few different ways of doing this. The easiest way is to note that 𝑥 remains constant at one and 𝑦 remains constant at two. However, 𝑧 can take any value we want. So as we move along our line, our value of 𝑥 doesn’t change and our value of 𝑦 doesn’t change. So the components of 𝑥 and 𝑦 in its direction vector will be zero. We can write that 𝐝 sub one is the vector zero, zero, one.
And of course, we could take any scalar multiple of this vector to be the direction vector of this line. We can do exactly the same thing to determine the direction vector of the second line. This time, our values of 𝑦 and 𝑧 remain constant at negative one and zero, respectively. So only the value of 𝑥 changes. This means the direction vector 𝐝 sub two of this line is the vector one, zero, zero.
And at this point, there’s many different ways we can answer this question. For example, we could calculate the dot product of these two vectors and their magnitudes and substitute them into the formula. This would work since we can calculate the dot product of the two vectors by taking the sum of the products of the corresponding components. We get zero times one plus zero times zero plus one time zero, which is just equal to zero.
We then see the numerator of the right-hand side of the equation is just zero. So we don’t need to calculate the magnitude of the two vectors, although we can see that these have magnitude one anyway. Therefore, since the right-hand side of this equation has numerator zero, we can just write this as the cos of 𝜃 is zero. We can then solve this equation for 𝜃 by taking the inverse cosine of both sides of the equation. And since the inverse cos of zero is 90 degrees, we can conclude that 𝜃 is 90 degrees.
It’s worth noting this is not the only method we could have used to answer this question. For example, we can notice something interesting about our two direction vectors. Since our first straight line only has its 𝑧-coordinate varying, it runs parallel to the 𝑧-axis. Similarly, since our second straight line only has its 𝑥-coordinate varying, it runs parallel to the 𝑥-axis. So we can choose the directional vectors of these two straight lines to be the unit directional vectors 𝐤 and 𝐢.
So the angle between the two straight lines is the same as the angle between these two directional vectors. And of course, we know the angle between these two directional vectors is 90 degrees, since it will be the same as the angle between any two of the axes.
Using either method, we can show the measure of the angle between the straight line 𝑥 is equal to one, 𝑦 is equal to two and the straight line 𝑦 is equal to negative one, 𝑧 is equal to zero is 90 degrees.