VG2
VG3
Lineær algebra
Interactive vectors
Adjust the components of vectors a and b with the sliders. See vector addition using the parallelogram rule, and explore the dot product and the angle between them.
Key formulas:
a + b = (a₁+b₁, a₂+b₂) | a · b = a₁b₁ + a₂b₂ | cos θ = (a · b) / (|a| · |b|)
When the dot product equals zero, the vectors are perpendicular (orthogonal).
VECTOR a (blue)
a₁ = 3.0
a₂ = 2.0
VECTOR b (orange)
b₁ = 1.0
b₂ = -2.0
|a| length
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|b| length
–
|a+b| length
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a · b dot prod
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θ angle
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Vector rules
a + b = (a₁+b₁, a₂+b₂)
Addition
a - b = (a₁-b₁, a₂-b₂)
Subtraction
|a| = √(a₁² + a₂²)
Length
a · b = a₁b₁ + a₂b₂
Dot product
cos θ = (a·b)/(|a||b|)
Angle
a · b = 0 → perpendicular
Orthogonality
k·a = (k·a₁, k·a₂)
Scalar mult.
a · a = |a|²
Self dot prod.
Key insight:
When a · b = 0 the vectors are perpendicular — the angle between them is exactly 90°. Try to make this happen with the sliders!
Vectors are the poetry of mathematics — direction and magnitude united in one symbol.
— Hermann Grassmann (1809–1877)