Video Transcript
Given that a straight line passes through the origin and the point four, one, two, determine the exact value of the cos of 𝜃 sub 𝑧. Note that 𝜃 sub 𝑧 is the measure of the angle between the straight line and the positive direction of the 𝑧-axis.
In this question, we’re told that a straight line passes through two points, the origin and the point with coordinates four, one, two. We need to use this information to determine the exact value of the cosine of the angle that this straight line makes at the positive direction of the 𝑧-axis.
Since the positive direction of the 𝑧-axis is a straight line in space, and so is the straight line passing through the origin and the point four, one, two, let’s start by recalling our formula for determining the angle between two straight lines in space. We recall if we have a line 𝐿 sub one with direction vector 𝐝 sub one and a line 𝐿 sub two with direction vector 𝐝 sub two, then the cos of 𝜃 will be equal to the dot product between 𝐝 sub one and 𝐝 sub two divided by the magnitude of 𝐝 sub one times the magnitude of 𝐝 sub two, where 𝜃 is the angle between the two straight lines 𝐿 sub one and 𝐿 sub two. And in this question, we’re asked to find the cos of 𝜃 sub 𝑧, which is the angle between two straight lines. So we can do this by finding the direction vectors of the two lines.
Let’s start by finding the direction vector of the first line. We’re told that this line passes through the origin and the point of coordinates four, one, two. We can then find the components of a direction vector of this line by noting one direction vector of this line will have terminal point with coordinates four, one, two and an initial point with coordinates zero, zero, zero. So one direction vector of this line will have components as the difference between the coordinates of each of these points. 𝐝 sub one is the vector four, one, two.
We can in fact do the exact same thing to find the direction vector of our second line. We just need to note the positive direction of the 𝑧-axis passes through the origin, and it also passes through the point with coordinates zero, zero, one. We can then use the exact same process to find a direction vector of this line. We get that 𝐝 sub two is equal to the vector zero, zero, one. We can then substitute these direction vectors 𝐝 sub one and 𝐝 sub two into this formula where 𝜃 will be 𝜃 sub 𝑧.
And although this would work, it’s usually easier to evaluate the numerator and denominator of the right-hand side of the equation separately. So let’s start by evaluating the numerator the dot product between vectors 𝐝 sub one and 𝐝 sub two. We need to determine the dot product of the vector four, one, two and the vector zero, zero, one. And to do this, we recall 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. In this case, that gives us four times zero plus one times zero plus two multiplied by one. And we can then evaluate this. The first two terms are zero, and two times one is just equal to two.
Let’s now determine the magnitude of the two vectors. Let’s start with the magnitude of 𝐝 sub one. Remember that’s the square root of the sum of the squares of its components. In this case, that’s the square root of four squared plus one squared plus two squared, which we can then evaluate. It’s equal to the square root of 21. We could apply the same process to determine the magnitude of vector 𝐝 sub two. However, we can notice something interesting. Vector 𝐝 sub two is equal to the unit vector 𝐤. And this is of course a unit vector. So we know its magnitude is one. The magnitude of vector 𝐝 sub two is one.
We can now substitute these values into our formula to determine an expression for the cos of 𝜃 sub 𝑧. To do this, let’s start by clearing some space but keep some of the useful information on screen. We can now substitute these values into our formula. We get the cos of 𝜃 sub 𝑧 is equal to two divided by root 21. And we could leave our answer like this. This is an exact value for the cos of 𝜃 sub 𝑧. However, we can simplify this further by rationalizing the denominator. We’ll multiply the numerator and denominator by root 21. And this then gives us our final answer.
If 𝜃 sub 𝑧 is the measure of the angle between the straight line passing through the origin and the point four, one, two and the positive direction of the 𝑧-axis, then the cos of 𝜃 sub 𝑧 is two root 21 divided by 21.
