Why the Line ax + by = 0 Passes Through the Point (−b, a)

Why the Line ax + by = 0 Passes Through the Point (−b, a)

In ℝ², the equation ax + by = 0 describes a line that is perpendicular to the vector (a, b). This article explains exactly why—and why that line always passes through the point (−b, a).

Why the Line ax + by = 0 Passes Through the Point (−b, a)

1. Start with the Vector (a, b)

Consider the vector (a, b). To find a line perpendicular to it, we need a vector whose dot product with (a, b) is zero.

Try the vector (−b, a):

(a, b) · (−b, a) = a(−b) + b(a) = −ab + ab = 0

Therefore, (−b, a) is perpendicular to (a, b).

2. Any Scalar Multiple Also Works

If (−b, a) is perpendicular to (a, b), then any multiple λ(−b, a) is also perpendicular:

(a, b) · [λ(−b, a)] = λ[(a, b) · (−b, a)] = λ · 0 = 0

Let this perpendicular vector be (x, y). Then

(x, y) = λ(−b, a)

Every point on the line comes from a particular choice of λ.

3. Converting to an Equation

Since (x, y) is perpendicular to (a, b), we have:

(a, b) · (x, y) = 0

Expanding the dot product gives:

ax + by = 0

Thus the equation ax + by = 0 represents the full set of vectors perpendicular to (a, b).

4. Why the Line Passes Through (−b, a)

Set λ = 1 in (x, y) = λ(−b, a), giving the point:

(x, y) = (−b, a)

Substitute into the equation:

a(−b) + b(a) = −ab + ab = 0

So the point (−b, a) lies on the line.

Conclusion

The equation ax + by = 0:

  • is perpendicular to the vector (a, b),
  • contains every multiple of the perpendicular vector (−b, a),
  • and always passes through the point (−b, a).

Geometrically, this shows how a single equation encodes a direction and a perpendicular relationship in ℝ².

Personalised notes based on FP2/FP3 vector geometry. For educational use.

Comments

Post a Comment

Popular posts from this blog

The Method of Differences — A Clean Proof of the Sum of Cubes

The Method of Differences — A Clean Proof of the Sum of Cubes The method of differences is a remarkably elegant tool for evaluating finite sums. When each term of a series can be written in the form f(r+1) − f(r) , the sum “collapses” — all interior terms cancel, leaving only a boundary expression. This behaviour is called a telescoping sum . 1) Telescoping Sums Assume the general term u r can be written as: u r = f(r+1) − f(r). Then the finite sum from r = 1 to r = n becomes: Σ u r = Σ ( f(r+1) − f(r) ). To see what happens, write out a few terms: u₁ = f(2) − f(1) u₂ = f(3) − f(2) u₃ = f(4) − f(3) ⋮ uₙ = f(n+1) − f(n) When these are added, everything cancels except the first and last pieces: Σ u r = f(n+1) − f(1). This is the essence of the method: interior structure disappears, leaving just the difference between the final and initial states. 2) A Classic Application — The Sum of Cubes We will use this technique to prove the well-known ...
Read more

2×2 Orthogonal Matrix Mastery — A Generalised Construction

Image
2×2 Orthogonal Matrix Mastery — A Generalised Construction Orthogonal matrices in two dimensions reveal one of the cleanest structures in linear algebra. A 2×2 matrix is orthogonal when its columns (and rows) satisfy two conditions: They are perpendicular (their dot product is zero); They have unit length (their magnitude is one). This article presents a clear generalisation: any pair of perpendicular vectors with equal magnitude can be normalised to form an orthogonal matrix. 1. Begin with two perpendicular vectors Let the first vector be: (a, b) A perpendicular vector can be chosen as: (−b, a) Their dot products confirm orthogonality: (a, b) · (−b, a) = −ab + ab = 0 (a, −b) · (b, a) = ab − ab = 0 2. Compute their shared magnitude Both vectors have the same length: |(a, b)| = √(a² + b²) We can therefore normalise each one by dividing by √(a² + b²). 3. Form the matrix using the normalised vectors Place the two normalised vectors...
Read more

The Maclaurin Series — A Clean Derivation

Image
The Maclaurin Series — A Clean Derivation Many smooth functions can be written as an infinite polynomial. When this expansion is centred at x = 0 , we obtain the Maclaurin series . This article derives the Maclaurin formula directly from repeated differentiation, showing precisely why the coefficients involve derivatives and factorials. 1) Begin with a General Power Series Suppose a function f(x) can be expressed as f(x) = a₀ + a₁x + a₂x² + a₃x³ + … + aᵣxʳ + … The constants aᵣ are real coefficients whose values we wish to determine. 2) Evaluate at x = 0 f(0) = a₀ so a₀ = f(0). 3) Differentiate Once f′(x) = a₁ + 2a₂x + 3a₃x² + … + r·aᵣxʳ⁻¹ + … Setting x = 0 eliminates all higher powers: f′(0) = a₁. Thus, a₁ = f′(0). 4) Differentiate Again f″(x) = 2·1·a₂ + 3·2·a₃x + … + r(r−1)aᵣxʳ⁻² + … Evaluate at x = 0 : f″(0) = 2! · a₂ Hence a₂ = f″(0) / 2!. 5) The General Pattern Differentiate repeatedly. After r differentiations, a...
Read more