Video Transcript
Determine, in vector form, the equation of the straight line that passes through the points negative five, negative five, three and negative three, negative four, four.
We begin by recalling that when we write the equation of a line in vector form, it will be of the form 𝐫 is equal to 𝐫 sub zero plus 𝑡 multiplied by 𝐝, where 𝐫 sub zero is the position vector of a given point on the line and 𝐝 is the direction vector. We can calculate this direction vector by subtracting the position vectors of two points that lie on the line.
In this question, we are given two points that lie on the line. We will let the point negative five, negative five, three have coordinates 𝑥 sub zero, 𝑦 sub zero, 𝑧 sub zero and the point with coordinates negative three, negative four, four be 𝑥 sub one, 𝑦 sub one, 𝑧 sub one. It is important to note that we could substitute these in either order.
To find the direction vector of our line, we subtract the vector negative five, negative five, three from the vector negative three, negative four, four. When subtracting vectors, we simply subtract the corresponding components. Negative three minus negative five is equal to two. Negative four minus negative five is equal to one. And four minus three is also equal to one. A direction vector that passes through the points negative five, negative five, three and negative three, negative four, four is two, one, one.
The equation of the straight line is therefore equal to negative five, negative five, three plus 𝑡 multiplied by two, one, one. As already mentioned, 𝐫 sub zero can be the position vector of any point on the line. From the information in this question, we could replace negative five, negative five, three with negative three, negative four, four. We could also have found a different direction vector by subtracting the two position vectors in the opposite order. This would’ve given us a direction vector of negative two, negative one, negative one, which is the inverse of the other direction vector. Any combination of these position vectors and direction vectors would be valid.
