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Math101learn.math101.caCross Product
The cross product creates a vector perpendicular to two three-dimensional vectors, with magnitude equal to their parallelogram area.
Unlike the dot product, the cross product returns a direction—one normal to the plane of its two input vectors.
Definition
For
the cross product is
The output is a three-dimensional vector.
Determinant mnemonic
It is often organized as
Expanding along the first row creates a minus sign in the middle component. Writing the component formula explicitly helps avoid losing that sign.
Worked example
Its dot product with the second input is also zero.
Direction and the right-hand rule
The vector $\vec u\times\vec v$ points perpendicular to both inputs. Curl the fingers of the right hand from $\vec u$ toward $\vec v$ through the smaller angle; the thumb gives the cross-product direction.
Reversing order reverses direction:
The cross product is not commutative.
Magnitude and area
If $\theta$ is the angle between the vectors,
This equals the area of the parallelogram formed by the vectors. The corresponding triangle area is half:
Parallel vectors
If nonzero vectors are parallel, $\theta=0$ or $\pi$, so $\sin\theta=0$ and
Thus a zero cross product tests parallelism, with the zero-vector case handled carefully.
Finding a plane normal
Two nonparallel direction vectors in a plane have a cross product perpendicular to the plane. This normal vector can produce a scalar plane equation.
If points $A,B,C$ define a plane, compute $\overrightarrow{AB}\times\overrightarrow{AC}$ to obtain a normal.
Torque
Torque from position vector $\vec r$ and force $\vec F$ is
Its magnitude measures turning effect, and its direction gives the rotation axis by the right-hand rule. Perpendicular force produces the greatest torque for fixed magnitudes.
Scalar triple product
The scalar
has absolute value equal to the volume of the parallelepiped spanned by the vectors. A zero value signals coplanarity.
Common mistakes
Returning a scalar. The cross product returns a vector.
Dropping the negative sign in the middle component. Use the explicit formula.
Assuming order does not matter. Reversing order negates the result.
Using cosine for cross-product magnitude. Area uses sine.
Giving both normal directions as different planes. Opposite normal vectors describe the same plane orientation.
Quick self-check
- Is the operation in three dimensions and the output a vector?
- Were components and the middle sign computed correctly?
- Does the result dot to zero with both inputs?
- Does the right-hand rule match the input order?
- Is magnitude being interpreted as parallelogram area or twice triangle area?
- Are units and geometric meaning stated?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find ⟨1, 2, 0⟩ × ⟨3, −1, 4⟩.
- First component: 2(4) − 0(−1) = 8.
- Second: 0(3) − 1(4) = −4.
- Third: 1(−1) − 2(3) = −7.
End of lesson
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