How do you find a vector with initial point and terminal point?

Vectors: Connecting the Dots (Literally!)

So, you’re diving into vectors, huh? These things are super important in math and physics because they tell you not just how much of something you have, but also which way it’s going. Think of it like this: a vector is like an arrow pointing from one spot to another. And figuring out the vector when you know those two spots? That’s what we’re cracking open today.

Basically, a vector is that arrow, right? The starting point is the “initial point,” and where it lands is the “terminal point.” But here’s the kicker: it’s not just the line itself. It’s the movement from the start to the end. Think of it as the “as the crow flies” route. It’s got a length (magnitude) and a direction, and that’s what makes it a vector.

Okay, so how do we actually find this vector? There’s a simple trick. A vector is just the difference between the coordinates of the end point and the start point. Imagine you’ve got point A at (x₁, y₁) and point B at (x₂, y₂). The vector AB? Here’s the magic formula:

AB = (x₂ – x₁, y₂ – y₁)

And if you’re working in 3D space, with points like A = (x₁, y₁, z₁) and B = (x₂, y₂, z₂), it’s just as easy:

AB = (x₂ – x₁, y₂ – y₁, z₂ – z₁)

That (x, y) or (x, y, z) thing you get at the end? Those are the vector’s components. They tell you how much the vector moves along each axis.

Let’s break it down step-by-step, nice and easy:

Quick Example:

Say A is at (2, 3) and B is at (5, 7). Then:

AB = (5 – 2, 7 – 3) = (3, 4)

So, this vector AB goes 3 units horizontally and 4 units vertically. Easy peasy.

Now, a little heads-up: don’t mix up the vector form with the coordinates of a point (x, y). The first one is about the movement, the second is about a location.

Oh, and there’s something called a “position vector.” It’s just a vector that starts at the origin (0, 0). So, its components are the same as the coordinates of its end point. You can think of any vector as the difference between two position vectors. It’s like saying AB = OB – OA. Clever, right?

One more thing: order matters. If you subtract the wrong way, you get the vector from B to A, which points in the opposite direction.

Why bother with all this? Well, vectors pop up everywhere:

• Physics: Figuring out how far something moved and in what direction.
• Graphics: Making things move around on a screen.
• Engineering: Checking if bridges are gonna fall down (okay, it’s more complicated than that, but you get the idea!).
• Navigation: Plotting courses for ships and planes.

So, there you have it. Finding a vector from two points is all about subtracting coordinates. Get this down, and you’re well on your way to mastering vectors and all the cool stuff they let you do. It might seem a bit abstract now, but trust me, it’s a seriously useful tool in your math and science toolbox.