Question Video: Finding the Unknown Coefficients in the Equations of Two Straight Lines in Three Dimensions Given That They Are Perpendicular Mathematics

Video Transcript

If the two straight lines 𝑥 plus three all divided by negative eight is equal to 𝑦 minus four all divided by four 𝑛 is equal to 𝑧 plus one all divided by 10 and 𝑥 plus five all divided by four 𝑛 is equal to 𝑦 plus 10 all divided by negative four is equal to 𝑧 minus three all divided by negative five are perpendicular, find 𝑛.

In this question, we’re given two straight lines, and we’re told that these two straight lines are perpendicular. We need to use this information to find the value of 𝑛. And to do this, let’s start by looking at the forms we’re given our two straight lines in. The’re given as three equal expressions in a very useful form. And this form of straight lines has a name. It’s called the symmetric form. This is when we’re given our straight line in the form 𝑥 minus 𝑥 zero all divided by 𝑎 is equal to 𝑦 minus 𝑦 zero all divided by 𝑏 is equal to 𝑧 minus 𝑧 zero all divided by 𝑐 for some nonzero constants 𝑎, 𝑏, and 𝑐.

Now we want to find the value of 𝑛. And to do this, we’re going to need to use the fact that our two straight lines are perpendicular. We know that the symmetric form is a useful form for straight lines. However, it’s difficult to use the fact that these lines are perpendicular in this form. It would be easier if we had this in a form which told us the directions our line were facing, for example, the vector form. And in fact, it’s very easy to convert between the two.

For example, if we’re given the symmetric form of a straight line, we can write the vector form as the following. 𝑟 will be equal to the vector 𝑥 zero, 𝑦 zero, 𝑧 zero plus 𝑡 times the vector 𝑎, 𝑏, 𝑐. In other words, we know that our vector will pass through the point 𝑥 zero, 𝑦 zero, 𝑧 zero, and we know that our line will point in the direction of the vector 𝑎, 𝑏, 𝑐. So let’s use this information to write both of the straight lines given to us in the question in symmetric form into vector form.

Let’s start with the first straight line. We can see we’re adding three to our value of 𝑥. This is the same as subtracting negative three. So our value of 𝑥 zero is negative three. Next, we can see we’re subtracting four from 𝑦. So our value of 𝑦 zero is going to be equal to four. Finally, we see we’re adding one to our value of 𝑧. This is of course the same as subtracting negative one. So our value of 𝑧 zero is going to be negative one. And our values of 𝑎, 𝑏, and 𝑐 are just going to be the denominators of our fraction. 𝑎 is negative eight, 𝑏 is four 𝑛, and 𝑐 is 10. So we were able to find the vector form of the equation of our first straight line. Let’s do the same for our second one.

We do this in exactly the same way. We see that 𝑥 zero will be native five, 𝑦 zero will be negative 10, and 𝑧 zero will be three. And once again, our values of 𝑎, 𝑏, and 𝑐 will be the denominators of our fractions. 𝑎 is four 𝑛, 𝑏 is negative four, and 𝑐 is negative five. So now we have the vector forms of both of our straight lines. Now, we need to recall what it means for two straight lines to be perpendicular in their vector forms.

Remember, two straight lines being perpendicular means that they cross at right angles. So what does this mean for the vector forms of our equations? This means that their direction vectors must be perpendicular. And we know how to check if two vectors are perpendicular. They’ll be perpendicular if their scalar product is equal to zero. Therefore, to check that these two lines are perpendicular, we need to check whether the scalar product of their direction vectors is equal to zero.

So let’s calculate the scalar product or dot product of these two direction vectors. Remember, to do this, we need to multiply the corresponding components of each vector together and then add all of these together. So by evaluating the scalar product of these two vectors, we get negative eight times four 𝑛 plus four 𝑛 times negative four plus 10 multiplied by negative five. And if we simplify this expression, we see that it’s equal to negative 48𝑛 minus 50.

But remember, in the question, we’re told that these two lines are perpendicular. And remember, we know that if these two lines are perpendicular, then the scalar product of these direction vectors must be equal to zero. So, in other words, we must have that negative 48𝑛 minus 50 is equal to zero. We can then solve this equation for our value of 𝑛. And we can rearrange this equation for 𝑛. We add 50 to both sides and then divide through by negative 48. This gives us that 𝑛 is equal to negative 25 divided by 24.

Therefore, we were able to show if the two straight lines 𝑥 plus three all divided by negative eight is equal to 𝑦 minus four all divided by four 𝑛 is equal to 𝑧 plus one all divided by 10 and 𝑥 plus five all divided by four 𝑛 is equal to 𝑦 plus 10 all divided by negative four is equal to 𝑧 minus three all divided by negative five are perpendicular, then our value of 𝑛 must be equal to negative 25 divided by 24.