The cross product of two vectors is a third vector perpendicular to both; it exists only in three dimensions. Its magnitude is the area of the parallelogram the two span. A plane is fixed by a point and a normal vector ⟨a, b, c⟩: a(x − x₀) + b(y − y₀) + c(z − z₀) = 0. The coefficients are the normal.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Computing a cross product
u × v = ⟨u₂v₃ − u₃v₂, u₃v₁ − u₁v₃, u₁v₂ − u₂v₁⟩. For ⟨1, 0, 0⟩ × ⟨0, 1, 0⟩: ⟨0 − 0, 0 − 0, 1 − 0⟩ = ⟨0, 0, 1⟩. i × j = k, as the right hand says.
A worked one
u = ⟨1, 2, 3⟩, v = ⟨4, 5, 6⟩: ⟨2·6 − 3·5, 3·4 − 1·6, 1·5 − 2·4⟩ = ⟨−3, 6, −3⟩. Check perpendicularity: ⟨−3, 6, −3⟩ · ⟨1, 2, 3⟩ = −3 + 12 − 9 = 0. And · v: −12 + 30 − 18 = 0.
Order matters
v × u = ⟨3, −6, 3⟩, the exact opposite of u × v. Swapping the order reverses the result, which never happens with a dot product. Parallel vectors give the zero vector.
Its length is an area
|⟨−3, 6, −3⟩| = √(9 + 36 + 9) = √54 ≈ 7.35: the area of the parallelogram spanned by u and v. Half of that, 3.67, is the area of the triangle with those two sides.
A plane through three points
P(1, 0, 0), Q(0, 2, 0), R(0, 0, 3): PQ = ⟨−1, 2, 0⟩, PR = ⟨−1, 0, 3⟩. PQ × PR = ⟨6, 3, 2⟩, the normal. Plane: 6(x − 1) + 3y + 2z = 0, so 6x + 3y + 2z = 6. Check: Q gives 6, R gives 6.
Reading a plane equation
3x − 2y + z = 7 has normal ⟨3, −2, 1⟩, already written down. Two planes are parallel when their normals are parallel; the angle between planes is the angle between their normals.
Step 2: Try It Yourself
Tap and try it out.
Find the plane through P(1, 0, 0), Q(0, 2, 0) and R(0, 0, 3), and the area of triangle PQR
- 1PQ = ⟨−1, 2, 0⟩, PR = ⟨−1, 0, 3⟩end minus start, twice
Step 0 of 4
- 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: In Real Life
A wrench and a roof
Torque is the cross product of the wrench’s arm and the force: a vector along the bolt, whose size is the turning effect. Three points on a roof define its plane through a cross product.
Step 4: 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 and substitutes: 2(x − 1) − 1(y − 2) + 4(z − 3) = 0.
Step 5: 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⟩ meters. What is its area in square meters?
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?
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.
Step 6: 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, reverses when the order swaps, and exists only in three dimensions.
- Its magnitude is the area of the parallelogram the two vectors span; half of it is a triangle.
- A plane is fixed by a point and a normal, and the normal’s components become the coefficients.
- Three points give a plane: two edge vectors, their cross product, then the point-normal form.