# Justin's Blog

## February 18, 2012

### Basis Vector

• A normalized vector that defines a "basis" for a coordinate scheme. Basis vectors are orthogonal to each other, and they are often written with a "hat." (Like this: $\hat{i}$)

### Normalized Vector

• A vector with a length of one. A normalized vector can be applied to a scalar to give it direction.

### Orthogonal Vectors

• When the dot product of two vectors is zero, they are said to be orthogonal. For example, two perpendicular lines are orthogonal.

There are as many basis vectors as there are dimenions in a coordinate scheme.

Thus: a 3D rectangular scheme (called $ℝ^{3}$ for Real, 3 dimensions) has 3 basis vectors, all perpendicular to each other.

You often see i, j, and k used as basis vectors. These usually refer to $ℝ^{3}$

You can write them out in vector form. $$\hat{i} = (1, 0, 0)$$ $$\hat{j} = (0, 1, 0)$$ $$\hat{k} = (0,0,1)$$

They are useful because you can describe any point in a coordinate system as a sum of the basis vectors, multiplied by a number.

$$(3, 4, 5)$$

Means the same as... $$3\hat{ i} + 4\hat{j} + 5\hat{k}$$

It can make addition and multiplication conceptually simpler.