I know this should be very simple to understand but it makes no sense to me.
Why use for example 5i + 6j + 7k ?
And what is a unit vector compared to a normal vector? Online it says they’re used to “specify the direction of a vector” but I just don’t understand what’s meant by that.
In: 5
Vectors can be used as a way to ‘define’ space. For example, pick a corner in your room; that is the ‘origin’. You have two directions along the floor and one up along the walls. These are your ‘base’ vectors (let’s say **i**, **j**, **k**, respectively).
When you state the position **p** of an object in the room, they will be in terms of steps along those vectors; say, 5 steps along **i**, 6 along **j** and 7 along **k**. In short: **p** = 5**i** + 6**j** + 7**k**. That’s what that notation means.
The (5, 6, 7) you often see are just the coordinates: scalars for a certain set of vectors that define your space. Now usually, there’s a set of ‘default’ vectors (like x and y axis on a graph) so you don’t need to mention the vectors explicitly, but this is not always true. In the room example, you could have used a different corner. Or you could have used vectors that are not at 90° angles (like following a diamond-shaped floor pattern). In that case, you’d have a different set of base vectors: **i’**, **j’**, **k**’. If you follow the (5, 6, 7) along those vectors, you end up in a different place. If you want to be sure you’re not misunderstood, you need both the vectors and coordinates.
As for unit vectors: so far I’ve only really mentioned the direction of the vectors. But vectors have both a direction and magnitude. When I said ‘in terms of steps along those vectors’, it’s important to know how long those vectors are: if you used ‘1 meter’ long vectors, (5, 6, 7) would again end you in a different spot than if you used ‘1 foot’ long vectors.
A unit vector is simply a vector with a length of 1. When you do math with vectors, their length will often come up as a multiplier. Since 1*x is just x, it’s a very convenient multiplier. Effectively, you’ve taken the ‘magnitude’ out of the equation, so you’re just left with the ‘direction’ part.
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