Find the vector equation of the straight line passing through the origin and the point zero, four.
Let’s begin by recalling the general form for the vector equation of a straight line. It can be written in a number of ways. But one way is 𝐫 equals 𝐚 plus 𝑘 times 𝐝. The vector 𝐚 is a position vector of a point that this line passes through. 𝐝 is the direction vector of the line. And 𝑘, which is sometimes represented by a different letter, is simply a scalar. And so for our line, we need to find a point that it passes through and its direction vector.
In fact, we’re told two points that our line passes through. It passes through the origin — that’s the point with coordinates zero, zero — and the point zero, four. So we could choose either of these. Let’s choose the origin. Since the position vector of a point on the line is measured from the origin, we can say that the origin itself must have the position vector zero, zero. We move zero units in any direction to get from the origin to the origin. Now, whilst we could represent this with a column vector, we can also represent it with these angled brackets.
Next, we’re going to find the direction vector. And so we’re interested in the second point that our line passes through. Since we know the line passes between the point with position vector zero, zero and the point zero, four, which has position vector zero, four, its direction vector must be the difference of these. It must be zero, four minus zero, zero.
And, of course, to subtract two two-dimensional vectors, we simply subtract their components. Zero minus zero is zero, and four minus zero is four. And, once again, we can represent this with our column vector or our vector using these angled brackets.
We’re ready to replace our position vector and direction vector in our general equation. And so the vector equation of our straight line is 𝐫 equals zero, zero plus 𝑘 times zero, four. We, of course, don’t actually need to write the vector zero, zero. So we say that 𝐫 is equal to 𝑘 times zero, four.
Now, of course, we don’t need to calculate the value of 𝑘. It just tells us that when we’re moving along the line, we take multiples of the direction vector zero, four. 𝑘 itself is a scalar; it’s not a vector quantity.