Assuming that we have three-dimensional real vectors :

$Assumptions = (u | v | w) ∈ Vectors[3, Reals];

we can use e.g. various tensor functions (new in ver. 9) e.g. TensorReduce to reduce (simplify) a tensor expression, e.g.

TensorReduce[ v.v + w.w - (v + w).(v + w) ]
TensorReduce[u \[Cross] (v \[Cross] w) ]

-2 v.w

-w u.v + v u.w

We can do more interesting reductions, let's show e.g. the Jacobi identity:

TensorReduce[ u \[Cross] (v \[Cross] w) + v \[Cross] (w \[Cross] u) + 
              w \[Cross] (u \[Cross] v) ]

0

or writing it in a traditional form:

Defer[   u \[Cross] (v \[Cross] w) + v \[Cross] (w \[Cross] u)
       + w \[Cross] (u \[Cross] v)] == 
TensorReduce[   u \[Cross] (v \[Cross] w) + v \[Cross] (w \[Cross] u)
              + w \[Cross] (u \[Cross] v) ] // TraditionalForm

Another common identities

TensorExpand[ (u \[Cross] v) \[Cross] (u \[Cross] w) ]
TensorExpand[ (u \[Cross] v).(u \[Cross] v) ]

u u \[Cross] v . w

-(u.v)^2 + u.u  v.v

Take a lok at new differential operators:

Curl[ Curl[ f[x, y, z], {x, y, z}], {x, y, z}] == Laplacian[ f[x, y, z], {x, y, z}]

True

Assuming that we have three-dimensional real vectors :

$Assumptions = (u | v | w) ∈ Vectors[3, Reals];

we can use e.g. various tensor functions (new in ver. 9) e.g. TensorReduce to reduce (simplify) a tensor expression, e.g.

TensorReduce[ v.v + w.w - (v + w).(v + w) ]
TensorReduce[u \[Cross] (v \[Cross] w) ]

-2 v.w

-w u.v + v u.w

We can perform more interesting reductions, let's show e.g. the Jacobi identity:

TensorReduce[ u \[Cross] (v \[Cross] w) + v \[Cross] (w \[Cross] u) + 
              w \[Cross] (u \[Cross] v) ]

0

or write it in a traditional form:

Defer[   u \[Cross] (v \[Cross] w) + v \[Cross] (w \[Cross] u)
       + w \[Cross] (u \[Cross] v)] == 
TensorReduce[   u \[Cross] (v \[Cross] w) + v \[Cross] (w \[Cross] u)
              + w \[Cross] (u \[Cross] v) ] // TraditionalForm

Another common identity

TensorExpand[ (u \[Cross] v) \[Cross] (u \[Cross] w) ]
TensorExpand[ (u \[Cross] v).(u \[Cross] v) ]

u u \[Cross] v . w

-(u.v)^2 + u.u  v.v

Take a look at new differential operators:

Curl[ Curl[ f[x, y, z], {x, y, z}], {x, y, z}] == Laplacian[ f[x, y, z], {x, y, z}]

True

Assuming that we have three-dimensional real vectors :

$Assumptions = (u | v | w) ∈ Vectors[3, Reals];

we can use e.g. various tensor functions (new in ver. 9) e.g. TensorReduce to reduce (simplify) a tensor expression, e.g.

TensorReduce[v.v + w.w - (v + w).(v + w)]
TensorReduce[u\[Cross](v\[Cross]w)]

-2 v.w

-w u.v + v u.w

We can do more interesting reductions, let's show e.g. the Jacobi identity:

TensorReduce[ u\[Cross](v\[Cross]w) + v\[Cross](w\[Cross]u) + w\[Cross](u\[Cross]v)]

0

Assuming that we have three-dimensional real vectors :

$Assumptions = (u | v | w) ∈ Vectors[3, Reals];

we can use e.g. various tensor functions (new in ver. 9) e.g. TensorReduce to reduce (simplify) a tensor expression, e.g.

TensorReduce[ v.v + w.w - (v + w).(v + w) ]
TensorReduce[u \[Cross] (v \[Cross] w) ]

-2 v.w

-w u.v + v u.w

We can do more interesting reductions, let's show e.g. the Jacobi identity:

TensorReduce[ u \[Cross] (v \[Cross] w) + v \[Cross] (w \[Cross] u) + 
              w \[Cross] (u \[Cross] v) ]

0

or writing it in a traditional form:

Defer[   u \[Cross] (v \[Cross] w) + v \[Cross] (w \[Cross] u)
       + w \[Cross] (u \[Cross] v)] == 
TensorReduce[   u \[Cross] (v \[Cross] w) + v \[Cross] (w \[Cross] u)
              + w \[Cross] (u \[Cross] v) ] // TraditionalForm

Another common identities

TensorExpand[ (u \[Cross] v) \[Cross] (u \[Cross] w) ]
TensorExpand[ (u \[Cross] v).(u \[Cross] v) ]

u u \[Cross] v . w

-(u.v)^2 + u.u  v.v

Take a lok at new differential operators:

Curl[ Curl[ f[x, y, z], {x, y, z}], {x, y, z}] == Laplacian[ f[x, y, z], {x, y, z}]

True