Question Video: Finding the Measure of the Angle between a Straight Line and a Coordinate Axis Mathematics • Third Year of Secondary School

Video Transcript

Find, to the nearest second, the measure of the angle between the straight line 𝑥 plus one over two is equal to 𝑦 minus two over negative four is equal to 𝑧 plus two over five and the positive direction of the 𝑥-axis.

In this question, we’re asked to determine the measure of the angle between two straight lines. And we need to give this angle to the nearest second. To do this, let’s start by recalling how we find the measure of the angle between two straight lines. We know if we have two straight lines with direction vectors 𝐝 sub one and 𝐝 sub two, then the angle 𝜃 between the two lines will satisfy the equation cos of 𝜃 is 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. Therefore, to find the angle between the two lines, we need to find the direction vector of each line.

Let’s start by finding the direction vector of the first line. We can do this by recalling if a straight line is given in Cartesian form, then the denominators of these fractions represents the components of its direction vector. And when using this method, it is important to check the signs of the variables. They all need to be positive. So, in this case, the direction vector 𝐝 sub one is the vector two, negative four, five. Our second straight line is just the positive direction of the 𝑥-axis. So the 𝑦- and 𝑧-coordinates of every point on this line remain constant at zero. Only the 𝑥-coordinate changes. So we’ll choose the unit directional vector 𝐢, which is the vector one, zero, zero, to be the direction vector of this line.

We could now substitute these vectors into our equation involving the angle between the two lines. However, it’s easier to evaluate both the numerator and denominator of the right-hand side of this equation first. Let’s start by finding the dot product of the two vectors. We can do this by recalling to find the dot product of two vectors of the same dimension, we just need to find the sum of the products of the corresponding components. However, in this case, our second vector has two components equal to zero. So, in this case, it’s just equal to the product of the first components of the two vectors. Two times one is two.

Next, we need to determine the magnitude of the two vectors. Let’s start with the magnitude of vector 𝐝 sub one. That’s the square root of the sum of the squares of its components. That’s the square root of two squared plus negative four squared plus five squared, which we can evaluate is root 45, which we could simplify to give three root five. And we could apply the same process to determine the magnitude of vector 𝐝 sub two. However, we chose this to be the unit vector 𝐢, so we already know its magnitude is one. We can now substitute these values into our equation. We get that cos of 𝜃 is equal to two divided by root 45 times one, which simplifies to root 45.

We can now solve for 𝜃 by taking the inverse cosine of both sides of the equation. This then gives us that 𝜃 is the inverse cos of two over root 45. And making sure our calculator is in degrees mode, we get 72.65 and this expansion continues degrees. But the question wants us to give our answer to the nearest second. So we’re going to need to convert this angle into degrees, minutes, and seconds. To do this, we start by noting there are 72 degrees in this angle. This leaves us with a remaining angle of 0.65 and this expansion continues degrees. Since there are 60 minutes in a degree, we can multiply this angle by 60 to convert it into minutes. Evaluating this and making sure we use the exact values, we get 39.23 and this expansion continues minutes.

So we can see that this angle has 39 minutes and then a remaining angle of 0.23 and this expansion continues minutes. And once again, we’re going to want to know what this remaining angle is in seconds. And since there are 60 seconds in a minute, we can do this by multiplying our angle by 60. And by using the exact angle, we get 14.16 and this expansion continues seconds. We want to give our answer to the nearest second. So we need to look at the first decimal digit to determine if we need to round up or round down. This is one, so we need to round down.

This then gives us our final answer. The measure of the angle between the straight line 𝑥 plus one over two is equal to 𝑦 minus two over negative four is equal to 𝑧 plus two over five and the positive direction of the 𝑥-axis to the nearest second is 72 degrees, 39 minutes, and 14 seconds.