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# Vector Analysis & Visualization

In the Wolfram Language, n-dimensional vectors are represented by lists of length n.

Calculate the dot product of two vectors:

 In[1]:= ⨯ `{1, 2, 3}.{a, b, c}`
 Out[1]=

Type ESCcrossESC for the cross product symbol:

 In[2]:= ⨯ `{1, 2, c}\[Cross]{a, b, c}`
 Out[2]=

Calculate a vector’s norm:

 In[1]:= ⨯ `Norm[{1, 1, 1}]`
 Out[1]=

Find the projection of a vector onto the x axis:

 In[2]:= ⨯ `Projection[{8, 6, 7}, {1, 0, 0}]`
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Find the angle between two vectors:

 In[3]:= ⨯ `VectorAngle[{1, 0}, {0, 1}]`
 Out[3]=

Calculate the gradient of a vector:

(For the `∇` symbol, use ESCgradESC.)
 In[1]:= ⨯ ```\!\( \*SubscriptBox[\(\[Del]\), \({x, y}\)]\({ \*SuperscriptBox[\(x\), \(2\)] + y, x + \*SuperscriptBox[\(y\), \(2\)]}\)\)```
 Out[1]=

Compute the divergence or curl of a vector field:

 In[2]:= ⨯ `Div[{f[x, y, z], g[x, y, z], h[x, y, z]}, {x, y, z}]`
 Out[2]=

The Wolfram Language has 2D and 3D functions suitable for visualizing vector fields:

 In[1]:= ⨯ `VectorPlot[{y, -x}, {x, -3, 3}, {y, -3, 3}]`
 Out[1]=
 In[2]:= ⨯ `VectorPlot3D[{y, -x, z}, {x, -3, 3}, {y, -3, 3}, {z, -3, 3}]`
 Out[2]=

Plot a vector field over a slice surface:

 In[3]:= ⨯ ```SliceVectorPlot3D[{y, -x, z}, "CenterPlanes", {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]```
 Out[3]=

QUICK REFERENCE: Vector Analysis `»`

QUICK REFERENCE: Vector Visualization `»`