### Video Transcript

Find the vector form of the equation of the straight line passing through the point 𝐴 two, five, five and parallel to the straight line passing through the two points 𝐵 negative three, negative two, negative six and 𝐶 five, zero, negative nine.

Alright, so here we have these three points in three-dimensional space. And we’re told that there’s a straight line passing through the points 𝐶 and 𝐵. And what we want to solve for is the equation of a straight line passing through point 𝐴 that runs parallel to the line through 𝐵 and 𝐶. Along with this, we want to express the equation of this line through point 𝐴 in what’s called vector form.

When we write a line in this form, it looks like this. We see there are two vectors in this equation. The first one is a vector from the origin of our coordinate frame to a point on the line with coordinates 𝑥 one, 𝑦 one, and 𝑧 one. And then from that point, we move up and down the line using this vector with components 𝑎, 𝑏, 𝑐 that’s parallel to it.

Here, the factor 𝑡 by which we multiply this parallel vector is called a scale factor. And to express all the points on the line, this value ranges across all possible numbers, positive, negative, and zero. To write the equation of a line in vector form then, we need to know a point that lies on it and a vector that’s parallel to it. We’ve been told that our line of interest does pass through point 𝐴. That means the only bit of information still missing is a vector parallel to our line.

We’re told though that the line that passes through points 𝐵 and 𝐶 is parallel to the line through 𝐴. And so using points 𝐵 and 𝐶, we can define a vector like this, we’ll call it 𝐯, that’s parallel to both lines. 𝐯 is equal to the vector form of the difference between the components of points 𝐵 and 𝐶. And carrying out this subtraction, we see that those components are eight, two, and negative three.

Now that we know a point that lies on our line and a vector parallel to it, we can write its equation in vector form. First, we write in a vector to our known point 𝐴. Then we add that to our scale factor 𝑡 times the vector parallel to our line.

It’s worth noting that this isn’t the only way to write the vector form of this line’s equation. For example, if we had subtracted points 𝐵 and 𝐶 in a different order, our vector 𝐯 would have different signs. But still, because of the values over which our scale factor 𝑡 ranges, the equation of that line would be the same as this line. That is, there would be different ways of writing the same line’s vector form.