Determine the vector form of the equation of the straight line passing through the point negative one, negative five, four and parallel to the vector negative three, five, one.
Alright, so here we have a point negative one, negative five, four and a vector with components negative three, five, one. And we imagine a line that passes through this point and is parallel to the given vector. Here, we want to solve for the vector form of the line’s equation. To start figuring this out, we can recall how we generally write the vector form of a line. It involves two vectors, one that goes from the origin of a coordinate frame to a known point on the line, we’ll say it has coordinates 𝑥 one, 𝑦 one, and 𝑧 one, and a second vector that is parallel to the line’s axis.
So then a point on the line and a vector parallel to the line are the two things we need to write the equation of a line in vector form. And we see that in this scenario, that’s exactly what we’re given. We can write then that 𝐫, the vector representing our line, equals a vector from the origin of our coordinate frame to our known point, negative one, negative five, four, plus a scale factor we’ll call 𝑡 multiplied by our vector that’s parallel to the line. This is the equation of our line in vector form.