# Vectors¶

A vector in Taichi can have two forms:

• as a temporary local variable. An `n` component vector consists of `n` scalar values.
• as an element of a global tensor. In this case, the tensor is an N-dimensional array of `n` component vectors.

In fact, `Vector` is simply an alias of `Matrix`, just with `m = 1`. See Matrices and Tensors and matrices for more details.

## Declaration¶

### As global tensors of vectors¶

`ti.``Vector`(n, dt, shape = None, offset = None)
Parameters: n – (scalar) the number of components in the vector dt – (DataType) data type of the components shape – (optional, scalar or tuple) shape the tensor of vectors, see Tensors and matrices offset – (optional, scalar or tuple) see Coordinate offsets

For example, this creates a 5x4 tensor of 3 component vectors:

```# Python-scope
a = ti.Vector(3, dt=ti.f32, shape=(5, 4))
```

Note

In Python-scope, `ti.var` declares Tensors of scalars, while `ti.Vector` declares tensors of vectors.

### As a temporary local variable¶

`ti.``Vector`([x, y, ...])
Parameters: x – (scalar) the first component of the vector y – (scalar) the second component of the vector

For example, this creates a 3D vector with components (2, 3, 4):

```# Taichi-scope
a = ti.Vector([2, 3, 4])
```

## Accessing components¶

### As global tensors of vectors¶

`a[p, q, ...][i]`
Parameters: a – (Vector) the vector p – (scalar) index of the first tensor dimension q – (scalar) index of the second tensor dimension i – (scalar) index of the vector component

This extracts the first component of vector `a[6, 3]`:

```x = a[6, 3]

# or
vec = a[6, 3]
x = vec
```

Note

Always use two pairs of square brackets to access scalar elements from tensors of vectors.

• The indices in the first pair of brackets locate the vector inside the tensor of vectors;
• The indices in the second pair of brackets locate the scalar element inside the vector.

For 0-D tensors of vectors, indices in the first pair of brackets should be `[None]`.

### As a temporary local variable¶

`a[i]`
Parameters: a – (Vector) the vector i – (scalar) index of the component

For example, this extracts the first component of vector `a`:

```x = a
```

This sets the second component of `a` to 4:

```a = 4
```

TODO: add descriptions about `a(i, j)`

## Methods¶

`a.``norm`(eps = 0)
Parameters: a – (Vector) eps – (optional, scalar) a safe-guard value for `sqrt`, usually 0. See the note below. (scalar) the magnitude / length / norm of vector

For example,

```a = ti.Vector([3, 4])
a.norm() # sqrt(3*3 + 4*4 + 0) = 5
```

`a.norm(eps)` is equivalent to `ti.sqrt(a.dot(a) + eps)`

Note

Set `eps = 1e-5` for example, to safe guard the operator’s gradient on zero vectors during differentiable programming.

`a.``dot`(b)
Parameters: a – (Vector) b – (Vector) (scalar) the dot (inner) product of `a` and `b`

E.g.,

```a = ti.Vector([1, 3])
b = ti.Vector([2, 4])
a.dot(b) # 1*2 + 3*4 = 14
```
`a.``cross`(b)
Parameters: a – (Vector, 2 or 3 components) b – (Vector of the same size as a) (scalar (for 2D inputs), or 3D Vector (for 3D inputs)) the cross product of `a` and `b`

We use a right-handed coordinate system. E.g.,

```a = ti.Vector([1, 2, 3])
b = ti.Vector([4, 5, 6])
c = ti.cross(a, b)
# c = [2*6 - 5*3, 4*3 - 1*6, 1*5 - 4*2] = [-3, 6, -3]

p = ti.Vector([1, 2])
q = ti.Vector([4, 5])
r = ti.cross(a, b)
# r = 1*5 - 4*2 = -3
```
`a.``outer_product`(b)
Parameters: a – (Vector) b – (Vector) (Matrix) the outer product of `a` and `b`

E.g.,

```a = ti.Vector([1, 2])
b = ti.Vector([4, 5, 6])
c = ti.outer_product(a, b) # NOTE: c[i, j] = a[i] * b[j]
# c = [[1*4, 1*5, 1*6], [2*4, 2*5, 2*6]]
```

Note

This have no common with `ti.cross`. `a` and `b` do not have to be 3 or 2 component vectors.

`a.``cast`(dt)
Parameters: a – (Vector) dt – (DataType) (Vector) vector with all components of `a` casted into type `dt`

E.g.,

```# Taichi-scope
a = ti.Vector([1.6, 2.3])
a.cast(ti.i32) # [2, 3]
```

Note

Vectors are special matrices with only 1 column. In fact, `ti.Vector` is just an alias of `ti.Matrix`.

TODO: add element wise operations docs