Swatchai's Web Adventure: เมษายน 2026

วันจันทร์ที่ 27 เมษายน พ.ศ. 2569

The line of intersection of two planes using vectors (Part 2)

There’s a clean, fully vector-based formula for a point on the intersection line, with no elimination at all.

Setup

Two planes:

$\mathbf{n}_1 \cdot \mathbf{r} = d_1,\quad \mathbf{n}_2 \cdot \mathbf{r} = d_2$

Let:

n₁, n₂ = normal vectors
d = n₁ × n₂ = direction of the line

Key idea

The point ( r₀ ) lies:

in both planes
perpendicular to the direction ( d )

So we construct it directly using cross products.

Formula for the point

$\mathbf{r}_0 = \frac{ d_1 (\mathbf{n}_2 \times \mathbf{d}) + d_2 (\mathbf{d} \times \mathbf{n}_1) }{ |\mathbf{d}|^2 }$

Why this works (intuition)

( n₂×d ) lies in plane 2 and points toward satisfying plane 1
( d×n₁ ) lies in plane 1 and points toward satisfying plane 2
The combination balances both constraints exactly
Dividing by |d|² normalizes the result

Example

Planes:

	x + y + z = 1,  x - y + z = 3

Step 1: normals

	n₁ = (1, 1, 1),
	n₂ = (1, -1, 1)
    
( d₁ = 1, d₂ = 3 )

Step 2: direction

	d = n₁ × n₂ = (2, 0, -2)

Step 3: compute pieces

	n₂ × d = (1, -1, 1) × (2, 0, -2) = (2, 4, 2)

	d × n₁ = (2, 0, -2) × (1, 1, 1) = (2, -4, 2)

Step 4: plug into formula

$\mathbf{r}_0 = \frac{ 1(2,4,2) + 3(2,-4,2) }{ (2)^2 + 0^2 + (-2)^2 }$

	r₀ = (1, -1, 1)

Final line

Direction: (1, 0, -1)

	r(t) = (1, -1, 1) + t (1, 0, -1)

Takeaway

No solving equations
No choosing variables
Pure vector operations: cross products + scaling

วันอาทิตย์ที่ 26 เมษายน พ.ศ. 2569

The line of intersection of two planes using vectors (Part 1)

To find the line of intersection of two planes using vectors, you combine two key ideas:

The direction of the line comes from the cross product of the planes’ normals
A point on the line is found by solving the two plane equations together

Step 1: Write the planes

Suppose the planes are:

$\text{Plane 1: } a_1 x + b_1 y + c_1 z = d_1$

$\text{Plane 2: } a_2 x + b_2 y + c_2 z = d_2$

Their normal vectors are:

$\mathbf{n}_1 = (a_1, b_1, c_1), \quad \mathbf{n}_2 = (a_2, b_2, c_2)$

Step 2: Direction of the intersection line

The direction vector d is perpendicular to both normals:

$\mathbf{d} = \mathbf{n}_1 \times \mathbf{n}_2$

If this cross product is zero → planes are parallel (no intersection or infinite overlap).

Step 3: Find a point on the line

To get a specific point:

Solve the two plane equations simultaneously
Usually set one variable (like (z = 0)) and solve for (x, y)

Step 4: Write the vector equation of the line

Once you have:

A point $\mathbf{r}_0 = (x_0, y_0, z_0)$
Direction $\mathbf{d}$

The line is:

$\mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{d}$

Example

Find the intersection of:

    x + y + z = 1
    x - y + z = 3

1. Normals

$\mathbf{n}_1 = (1,1,1), \quad \mathbf{n}_2 = (1,-1,1)$

2. Direction

$\mathbf{d} = \mathbf{n}_1 \times \mathbf{n}_2 = (2, 0, -2)$

(Simplify → (1,0,-1))

3. Find a point

Set (z = 0):

    x + y = 1
    x - y = 3

Add:

$2x = 4 \Rightarrow x = 2$

Then:

	y = -1

So point = (2, -1, 0)

4. Final line

$\mathbf{r}(t) = (2,-1,0) + t(1,0,-1)$

Final takeaway

Cross product → gives direction
Solve equations → gives point
Combine → gives line

วันศุกร์ที่ 17 เมษายน พ.ศ. 2569

Vector's direction cosines and spherical earth model (Part 2)

In a spherical Earth model, the direction cosines of a position vector are essentially the components of the unit vector that points to a specific latitude (φ) and longitude (λ). [1]

The Coordinate Mapping

To relate these, we use an Earth-Centered, Earth-Fixed (ECEF) Cartesian system where:

x-axis: Points toward the Prime Meridian (0° longitude) at the Equator.
y-axis: Points toward 90° East longitude at the Equator.
z-axis: Points toward the North Pole (90° North latitude). [2, 3]

Conversion Formulas

If a point on Earth has latitude φ and longitude λ, its direction cosines (l, m, n) are calculated as:

Component along the x-axis
$l = \cos \alpha = \cos \phi \cos \lambda$
Component along the y-axis
$m = \cos \beta = \cos \phi \sin \lambda$
Component along the z-axis
$n = \cos \gamma = \sin \phi$
[1, 4, 5]

Key Differences from Standard Physics

It is important to note that "standard" spherical coordinates (ρ, θ, φ) used in math/physics differ from geographic coordinates: [1]

Latitude (φ): Measured from the Equator (up/down). In standard spherical math, the "polar angle" is measured from the North Pole (downward).
Relationship: The polar angle θ in math is equal to (90° - φ) in geography. This is why the z-component uses sin φ for latitude instead of the usual cos θ. [1, 5, 6]

Inverse Relationship

If you already have the direction cosines

(l, m, n)

, you can find the geographic coordinates using:

Latitude:
$\phi = \arcsin(n)$
Longitude:
$\lambda = \operatorname{atan2}(m, l)$
[4, 7]

This vector-based approach is often called the n-vector representation and is used in navigation software because it avoids "singularities" (mathematical glitches) at the North and South Poles. [8]

[1] https://moodle2.units.it [2] https://vulms.vu.edu.pk [3] https://math.univ-lyon1.fr [4] https://via-technology.aero [5] https://www.youtube.com [6] https://en.wikipedia.org [7] https://www.movable-type.co.uk [8] https://en.wikipedia.org

Vector's direction cosines and spherical earth model (Part 1)

Direction cosines are the cosines of the angles that a vector makes with the positive x, y, and z axes. For a vector

$\vec{v} = \langle v_x, v_y, v_z \rangle$

in three-dimensional space, the direction cosines (often denoted as l, m, and n) are calculated by dividing each component of the vector by its magnitude. [1, 2, 3, 4]

Finding the Direction Cosines of a Vector | Formula ... Understand Unit Vectors and Direction Cosines of Vectors ...

Formulas and Calculations

The direction cosines relate to the direction angles α (with the x-axis), β (with the y-axis), and γ (with the z-axis) as follows: [5, 6, 7]

Calculate Magnitude: First, find the length of the vector:
$|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$
Determine Cosines:

$l = \cos \alpha = \frac{v_x}{|\vec{v}|}$

$m = \cos \beta = \frac{v_y}{|\vec{v}|}$

$n = \cos \gamma = \frac{v_z}{|\vec{v}|}$

[8, 9, 10]

Essential Properties

Unit Vector: The direction cosines of a vector are exactly the components of its corresponding unit vector.
Fundamental Identity: The sum of the squares of the direction cosines always equals one:
$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = l^2 + m^2 + n^2 = 1$
Direction Ratios: Any set of numbers proportional to the direction cosines are called direction ratios (often a, b, c). [2, 11, 12, 13, 14, 15]