Cross

Cross[a,b]

gives the vector cross product of a and b.

Details

If a and b are lists of length 3, corresponding to vectors in three dimensions, then Cross[a,b] is also a list of length 3.
Cross[a,b] can be entered in StandardForm and InputForm as ab, a cross b or ab. Note the difference between \[Cross] and \[Times].
Cross is antisymmetric, so that Cross[b,a] is -Cross[a,b]. »
Cross[{x,y}] gives the perpendicular vector {-y,x}.
In general, Cross[v₁,v₂,…,v_n-1] is a totally antisymmetric product which takes vectors of length n and yields a vector of length n that is orthogonal to all of the v_i.
Cross[v₁,v₂,…] gives the dual (Hodge star) of the wedge product of the v_i, viewed as one‐forms in n dimensions.

Examples

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Basic Examples (3)

The cross product of two vectors in three dimensions:

Visualize the two initial vectors, the plane they span in and the product:

The cross product of a single vector in two dimensions:

Visualize the two vectors:

Enter using cross:

Scope (9)

Find the cross product of machine-precision vectors:

Cross product of complex vectors:

Cross product of exact vectors:

The cross product of arbitrary-precision vectors:

Cross product of symbolic vectors:

Compute the cross product of QuantityArray vectors:

The QuantityArray structure is preserved:

Format the result:

The cross product of a single vector in two dimensions:

The result is perpendicular to the original vector:

Define two vectors in three dimensions:

Verify that Cross is antisymmetric:

Define three vectors in four dimensions:

Compute the cross product of the vectors:

Verify that the product is orthogonal to all three vectors:

Compute all possible product orders; each swap of two vectors merely changes the overall sign:

Applications (10)

Geometric Applications (5)

Find the normal to the plane spanned by two vectors:

Verify that the result is perpendicular to both inputs:

The equation for the plane:

Find a vector perpendicular to a vector in the plane:

Verify that u and v are perpendicular:

Find a vector orthogonal to n-1 vectors in n dimensions:

Find the area of the parallelogram defined by two vectors:

Compare with a direct computation using Area:

This can also be computed as , with the angle between the vectors:

Visualize the parallelogram:

The Frenet–Serret system encodes every space curve's properties in a vector basis and scalar functions. Consider the following curve:

Define the tangent, normal and binormal vectors in terms of cross products of the first two derivatives:

These three vectors define a right-handed, orthonormal basis for :

Compute the curvature, , and torsion, , which quantify how the curve bends:

Verify the answers using FrenetSerretSystem:

Visualize the curve and the associated moving basis, also called a frame:

Physical Applications (5)

Find the torque about the origin of a force straight down applied at the point :

Torque is given by the formula :

Find the angular momentum of a particle of mass , velocity and position about the origin:

Angular momentum is given by , with linear momentum equal to :

Find the magnetic force on the particle of charge and velocity moving through a magnetic field of in the positive direction:

Magnetic force is given by :

Use UnitSimplify get the expected unit Newtons, and MatrixForm to format the vector:

Consider a particle constrained to rotate at a fixed distance from the axis:

Define the angular velocity by means of a cross product:

Many properties can be expressed in terms of . The linear velocity equals :

The perpendicular or centripetal acceleration equals :

Since and are orthogonal, it is immediate that $TemplateBox[{{a, _, {(, perp, )}}}, Norm]=TemplateBox[{omega}, Norm] TemplateBox[{{r, ^, {(, ', )}}}, Norm]$ :

The well-known formula for centripetal acceleration, $TemplateBox[{{a, _, {(, perp, )}}}, Norm]=(TemplateBox[{{r, ^, {(, ', )}}}, Norm]^2)/(TemplateBox[{r}, Norm])=(TemplateBox[{v}, Norm]^2)/(TemplateBox[{r}, Norm])$ , also holds:

The derivative of is the angular velocity :

The acceleration parallel to the direction of motion, , equals :

Note that the linear acceleration equals the sum :

Cross products with respect to fixed three-dimensional vectors can be represented by matrix multiplication, which is useful in studying rotational motion. Construct the antisymmetric matrix representing the linear operator , where is an angular velocity about the axis:

Verify that the action of is the same as doing a cross product with :

The rotation matrix at time is the matrix exponential of times the previous matrix:

Verify using RotationMatrix:

The point at time zero will be at time :

The velocity of will be given by :

And the vector from the axis of rotation to is $(v(t)xomega)/(TemplateBox[{omega}, Norm]^2)$ :

Visualize this motion and the associated vectors:

Properties & Relations (10)

If u and v are linearly independent, u×v is nonzero and orthogonal to u and v:

If u and v are linearly dependent, u×v is zero:

For three-dimensional vectors, $TemplateBox[{{{u, _, 1}, x, {u, _, 2}}}, Norm]=TemplateBox[{{u, _, 1}}, Norm] TemplateBox[{{u, _, 2}}, Norm]sin(theta)$ , with the angle between and :