Vectors (Further Pure 1)
Introduced| English | Meaning |
|---|---|
| plane/pleɪn/ | A flat two-dimensional surface extending indefinitely in every direction within it. |
| skew/skjuː/ | Describing lines in three-dimensional space that are neither parallel nor intersecting and do not lie in one plane. |
| vector product/ˈvektə ˈprɒdʌkt/ | The cross product of two three-dimensional vectors, perpendicular to both with magnitude equal to their magnitudes times the sine of the angle between them. |
| normal vector/ˈnɔːml ˈvektə/ | normal vector |
| perpendicular/ˌpɜːpənˈdɪkjʊlə/ | Meeting at a right angle of 90 degrees. |
| parallelogram/ˌpærəˈleləɡræm/ | A quadrilateral with both pairs of opposite sides parallel. |
| anti-commutative/ˈænti ˈkɒmjuːtətɪv/ | anti-commutative |
The mathematics of 3D space
- How do you find the distance from a point to a plane 平面? The angle between two surfaces? The shortest path between two skew 异面 lines?
- Planes and the vector product 向量积 give you the tools to solve all these problems — essential for engineering, architecture, and computer graphics.
Equations of planes
- A plane can be written in several forms:
The normal vector 法向量 n is perpendicular 垂直 to the plane; the scalar-product form is n·r = d.
- Cartesian: $ax + by + cz = d$
- Vector: $\mathbf{r}\cdot\mathbf{n} = p$ (where $\mathbf{n}$ is the normal vector)
- Parametric: $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}$
Worked example. A plane has normal $\mathbf{n} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and passes through $(1, 0, 2)$. Equation: $2x - y + 3z = 2(1) - 0 + 3(2) = 8$.
Vectors
a · b = |a||b|cos θ
The dot product is zero exactly when two vectors are perpendicular.
Scalar and vector products can find the shortest distance between two skew lines.
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These products let you compute distances, angles and intersections in 3D, including skew-line distances.
A plane has equation 2x − y + 3z = 8. The normal vector has components (2, −1, ?). Find the missing component.
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The coefficients of x, y, z in ax + by + cz = d give the normal vector (a, b, c) = (2, −1, 3).
The vector (cross) product
- The vector product $\mathbf{a}\times\mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin\theta\,\hat{\mathbf{n}}$ gives a vector perpendicular to both.
- Its length equals the area of the parallelogram 平行四边形 spanned by $\mathbf{a}$ and $\mathbf{b}$.
Order matters. $\mathbf{a}\times\mathbf{b} = -\mathbf{b}\times\mathbf{a}$. The vector product is anti-commutative 反交换 — swapping the order reverses the direction.
Forces like wind and water are vectors, with both size and direction
The vector product a × b gives a vector that is:
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a × b is perpendicular to both vectors; its length is |a||b|sin θ.
The magnitude |a × b| equals the:
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|a × b| = |a||b|sin θ, which is the parallelogram's area.
The vector product is commutative: a × b = b × a.
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The vector product is anti-commutative: a × b = −(b × a). Swapping the order reverses the direction.
If |a| = 3, |b| = 4, and the angle between them is 30°, |a × b| = |a||b|sin θ. Find it.
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|a × b| = 3 × 4 × sin 30° = 12 × 0.5 = 6.
Applications
- With scalar and vector products: distances, angles, where lines and planes meet, and the shortest distance between skew lines.
- Distance from point to plane: $d = \dfrac{|ax_0 + by_0 + cz_0 - d|}{\sqrt{a^2 + b^2 + c^2}}$.
The vector product is perpendicular to both vectors; its length is the parallelogram area
Finding the normal
- The normal to a plane containing two vectors $\mathbf{b}$ and $\mathbf{c}$ is $\mathbf{n} = \mathbf{b} \times \mathbf{c}$.
- This is the quickest way to find the Cartesian equation from a parametric form.
You've got it
- a plane: $ax + by + cz = d$ or $\mathbf{r}\cdot\mathbf{n} = p$
- the vector product $\mathbf{a}\times\mathbf{b}$ is perpendicular to both; $|\mathbf{a}\times\mathbf{b}|$ = parallelogram area
- use it for distances, angles, intersections, and skew-line distances
Write from a cross-product magnitude of 30
- Two vectors have magnitudes 5 and 6, and the angle between them is 90 degrees.
- Find the magnitude of their vector product. Swapping the order reverses the product.
Two vectors have magnitudes 5 and 6 with an angle of 90 degrees between them. What is the magnitude of their vector product, and what happens if the order is swapped?
The magnitude of a cross product is the product of the magnitudes times the sine of the included angle. The product is anti-commutative.
What is the magnitude of the vector product?
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5 times 6 times 1 is 30. Other options — 6 (the lesson product for magnitudes 3 and 4 at 30 degrees); 11 (the sum of the magnitudes).