The cross product of two vectors is a third vector perpendicular to both. Unlike the dot product it only exists in three dimensions.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Which way it points
The right-hand rule decides. Point the fingers along u, curl toward v, and the thumb gives u × v.
Order matters
v × u is the exact opposite of u × v. Swapping the order reverses the result, which never happens with a dot product.
Its length is an area
The magnitude of u × v equals the area of the parallelogram the two vectors span. Parallel vectors span nothing, so their cross product is zero.
Normal vectors
A plane is fixed by one point on it and one vector perpendicular to it. That perpendicular vector is called the normal.
The plane equation
With normal ⟨a, b, c⟩ through (x₀, y₀, z₀), the plane is a(x − x₀) + b(y − y₀) + c(z − z₀) = 0. The normal components are the coefficients.
A vector perpendicular to two others
The cross product of two vectors is a third vector at right angles to both, with direction given by the right-hand rule. It exists only in three dimensions, unlike the dot product.
Its length is an area
|u × v| = |u||v| sin θ, which is the area of the parallelogram they span. Zero length means the vectors are parallel, so the cross product tests parallelism as the dot product tests perpendicularity.
Order reverses the direction
u × v = −(v × u). The cross product is anticommutative, which follows from the right-hand rule and is a genuine departure from ordinary multiplication.
It defines planes
A plane is fixed by a point and a normal vector, and the cross product of two vectors in the plane supplies that normal. That is the standard route from three points to a plane equation.
Step 2: Try It Yourself
Tap and try it out.
- i-hat lands on(3, 1)
- j-hat lands on(1, 2)
- Determinant5
The shaded parallelogram is the image of the unit square, and its area is 5. That is exactly what the determinant measures.
Step 3: Watch an Example
One step at a time.
Watch Priya Build a Plane
Priya wants the plane through (1, 2, 3) with normal ⟨2, −1, 4⟩.
- Step 1
She writes the pattern a(x − x₀) + b(y − y₀) + c(z − z₀) = 0.
Step 4: Your Turn
Practice makes it stick.
The Panel
Problem 1 of 2
Two edges of a flat panel are ⟨3, 0, 0⟩ and ⟨0, 4, 0⟩ metres. What is its area in square metres?
The Reading
Problem 2 of 2
The plane 5x + 2y − z = 9. What is the x-component of a normal vector?
Cross and Plane
1 of 8
i × j gives which unit vector? Enter 1 for i, 2 for j, 3 for k.
2 of 8
u × u for any vector u. What is its magnitude?
3 of 8
Two vectors of length 3 and 5 meet at 90 degrees. What is the cross product magnitude?
4 of 8
Two parallel vectors. What is the magnitude of their cross product?
5 of 8
The plane 3x − y + 2z = 7. What is the z-component of the normal?
6 of 8
A triangle spanned by two vectors with cross product magnitude 18. What is the triangle area?
7 of 8
Sort each operation by what it returns.
Tap something to move it.
- Empty
- Empty
8 of 8
j × i points opposite to k. Enter 1 if that is true, 0 if false.
Step 5: Quick Check
Show what you know.
Question 1 of 2
Two vectors of length 4 and 6 meet at 30 degrees. What is the cross product magnitude?
Question 2 of 2
What is a normal vector to a plane?
What You Learned
- The cross product gives a vector perpendicular to both inputs, and only exists in three dimensions.
- Its magnitude is the area of the parallelogram the two vectors span.
- A plane is fixed by a point and a normal vector, whose components become the coefficients.