Vector geometry, projections and oriented area
| English | 中文 | Pinyin |
|---|---|---|
| cross product/krɒs ˈprɒdʌkt/ | 叉积 | chā jī |
| orthogonal/ɔːˈθɒɡənl/ | 正交 | zhèng jiāo |
A decision before an answer
- Two paths can point in the same general direction, meet at right angles, or oppose one another. A dot product distinguishes these cases; a cross product measures the area they span.
- Your goal: Use dot products to classify angles and compute projections.
Read the relationship
- For nonzero real vectors u,v, u·v=|u||v| cos θ. A positive, zero or negative dot product corresponds to an acute, right or obtuse smaller angle. The scalar projection of v along u is (v·u)/|u|; its vector projection is ((v·u)/(u·u))u. The difference from this projection is orthogonal to u. Do not confuse a projected vector with its signed scalar component. The zero vector is orthogonal to every vector but has no defined angle direction.
- Calculate triangle area and orientation using a cross product.
Triangle vertices are (0,0,0),(2,0,0),(0,3,0). What is its area?
The edge cross product has magnitude 6. Triangle area is half the spanned parallelogram area.
Use the defining rule
- For three-dimensional vectors, u×v is perpendicular to both with length |u||v| sin θ. Coordinate calculation uses (u₂v₃−u₃v₂, u₃v₁−u₁v₃, u₁v₂−u₂v₁). This length is the parallelogram area, so a triangle from two edge vectors has half that area. Reversing their order reverses the cross product but preserves area. Build both edge vectors from the same vertex; crossing two unrelated position vectors generally measures the wrong triangle.
- Construct or rule out planar dot-product sign configurations.
What is the vector projection of (3,4) onto the line spanned by (1,0)?
Multiply (1,0) by the coefficient ((3,4)·(1,0))/((1,0)·(1,0))=3.
Check the conditions
- The plane through a point p with nonzero normal n has equation n·(x−p)=0. Distance from q to the plane is |n·(q−p)|/|n|; the denominator normalises the scale of the equation. A scalar triple product u·(v×w) gives signed parallelepiped volume; its absolute value is geometric volume. Zero triple product means dependence of the three edge vectors, not necessarily that each pair is perpendicular or parallel.
- Construct or rule out planar dot-product sign configurations.
With p=(0,0,0), q=(2,0,0), r=(0,3,0), the edges are u=(2,0,0), v=(0,3,0). Their cross product is (0,0,6), giving triangle area 3. For w=(3,4) and u=(1,0), the vector projection is (3,0) and the orthogonal remainder is (0,4). The plane 2x−y+2z=6 has normal length 3, so its distance from the origin is 6/3=2.
The distance from the origin to 2x−y+2z=6 is ____.
Divide the absolute constant 6 by the normal length √(4+1+4)=3.
Apply the task format
- For four planar vectors, there are six unordered dot products. The configuration e₁,−e₁,e₂,−e₂ has two negative products and four zeros; e₁,e₁,e₂,e₂ has two positive products and four zeros. Four nonzero vectors cannot have every pairwise dot product negative. Order their directions around the circle: each consecutive angular gap would have to exceed 90°, forcing the sum of four gaps above 360°. A zero vector cannot rescue a strict-negative requirement, because its dot products vanish.
- Construct or rule out planar dot-product sign configurations.
A cross-product magnitude is parallelogram area, so halve it for a triangle. Dot-product zero is an algebraic orthogonality statement even for a zero vector. Plane distance must divide by normal length.
Which answer fits this case?
Use dot products to classify angles and compute projections
Four nonzero vectors in R² can have all six unordered pairwise dot products negative.
Consecutive direction gaps would all exceed 90°, contradicting their total 360°. Zero vectors would produce zero products instead.
Keep the distinctions
- orthogonal 正交 — Having a zero dot product in a real inner-product space.
- cross product 叉积 — An oriented perpendicular vector in three dimensions whose magnitude is spanned parallelogram area.
- Use dot products to classify angles and compute projections.
- Calculate triangle area and orientation using a cross product.
- Construct or rule out planar dot-product sign configurations.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.