The Cross Product and Planes

A vector perpendicular to two others.

12 problems · about 24 minutes · N-VM.C.11, G-GPE.A.1

What this lesson teaches

The student computes cross products, uses them for area, and writes the equation of a plane.

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.

Straightforward practice. Get the method working first.

5 problems

Two vectors of length 4 and 6 meet at 30 degrees. What is the cross product magnitude?
Answer 12
Why 4 × 6 × 0.5 = 12.
What is a normal vector to a plane?
Answer A vector perpendicular to every vector lying in the plane.
Why It stands perpendicular to the plane.
i × j gives which unit vector? Enter 1 for i, 2 for j, 3 for k.
Answer 3
Why The right-hand rule.
u × u for any vector u. What is its magnitude?
Answer 0
Why A vector spans no area with itself.
Two vectors of length 3 and 5 meet at 90 degrees. What is the cross product magnitude?
Answer 15
Why |u||v| sin θ, and sin 90 is 1.

The same ideas with more to keep track of.

3 problems

Two parallel vectors. What is the magnitude of their cross product?
Answer 0
Why sin 0 is 0.
The plane 3x − y + 2z = 7. What is the z-component of the normal?
Answer 2
Why Read the coefficient of z.
A triangle spanned by two vectors with cross product magnitude 18. What is the triangle area?
Answer 9
Why A triangle is half a parallelogram.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each operation by what it returns.
Answer Returns a number: u · v, The magnitude of u × v · Returns a vector: u × v, u + v
Why A magnitude is always a number.
j × i points opposite to k. Enter 1 if that is true, 0 if false.
Answer 1
Why Swapping the order reverses the result.
The Panel: Two edges of a flat panel are ⟨3, 0, 0⟩ and ⟨0, 4, 0⟩ metres. What is its area in square metres?
Answer 12
Why 12 square metres.
The Reading: The plane 5x + 2y − z = 9. What is the x-component of a normal vector?
Answer 5
Why 5.