Is the vector i j/k a unit vector?

Is i + j + k a Unit Vector? Let’s Break It Down.

So, you’re wondering about unit vectors, eh? Specifically, whether that i + j + k vector qualifies. Unit vectors are kind of a big deal in vector math. Think of them as signposts – they tell you the direction something’s pointing, and they always have a length of exactly 1. We usually give them a little hat, like û, to show they’re special. But what about our vector, i + j + k? Is it unit material? Let’s find out.

First, a quick refresher. Remember those i, j, and k things? They’re like the basic building blocks for describing movement in 3D space. i points right along the x-axis, j goes straight up the y-axis, and k shoots out towards you on the z-axis. Each one’s a unit vector in its own right, with a length of 1. You can combine them to describe any vector, like saying “go 1 unit right, 1 unit up, and 1 unit forward.” That’s what i + j + k is telling us.

Okay, so how do we figure out the length (or magnitude) of i + j + k? There’s a formula for that: if you have a vector V = xi + yj + zk, its length is |V| = √(x² + y² + z²). It looks complicated, but it’s really just Pythagoras in three dimensions!

In our case, i + j + k is the same as 1i + 1j + 1k, so x = 1, y = 1, and z = 1. Let’s plug those in: |i + j + k| = √(1² + 1² + 1²) = √(1 + 1 + 1) = √3.

Aha! The length of i + j + k is √3, which is about 1.732. Definitely not 1. So, here’s the verdict: i + j + k is not a unit vector. Close, but no cigar.

Now, what if we wanted a unit vector pointing in the same direction as i + j + k? That’s where “normalizing” comes in. Normalizing just means shrinking or stretching a vector until its length is exactly 1, without changing its direction. To do it, you divide each part of the vector by its original length. So, our unit vector û would be (1/√3)i + (1/√3)j + (1/√3)k.

If you calculate the length of that vector, you’ll find it’s exactly 1. We’ve successfully created a unit vector that points the same way as i + j + k! Pretty neat, huh?