The Cross Product and Planes

A vector perpendicular to two others.

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

NameDate

Figure — use these to answer the problems

The shaded area is what a cross product measures. Make the two arrows parallel and watch it vanish.

u, x-component3

v, x-component1

u, y-component1

v, y-component2

Figure 1

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
2
What is a normal vector to a plane?
- A vector perpendicular to every vector lying in the plane.
- Any vector lying inside the plane.
i × j gives which unit vector? Enter 1 for i, 2 for j, 3 for k.
Answer
u × u for any vector u. What is its magnitude?
Answer
Two vectors of length 3 and 5 meet at 90 degrees. What is the cross product magnitude?
Answer

The same ideas with more to keep track of.

3 problems

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

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

4 problems

Sort each operation by what it returns.
Write each item under the heading it belongs to: u · v · u × v · The magnitude of u × v · u + v
Returns a number
Returns a vector
j × i points opposite to k. Enter 1 if that is true, 0 if false.
Answer
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
The Reading
The plane 5x + 2y − z = 9. What is the x-component of a normal vector?
Answer