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Scalar product
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The scalar (aka dot) product, and its connection to angles between vectors.
Want a deeper conceptual understanding? Try our interactive lesson!
Scalar product
AHL AI 3.13
The scalar product, also called the dot product, takes two vectors and produces a scalar (a number). For two vectors v=⎝⎛v1v2v3⎠⎞ and w=⎝⎛w1w2w3⎠⎞, the scalar product is calculated as:
v⋅w=v1w1+v2w2+v3w3📖
This operation combines corresponding components of each vector, resulting in a single numerical value.
The scalar product measures the extent to which two vectors point in the same direction. The bigger the vectors and the more they point in the same direction, the bigger the scalar product, since only the "parallel parts" contribute to the dot product. It's is a scalar quantity measuring the amount of similarity.
Angle between vectors
AHL AI 3.13
The scalar product also has a geometric interpretation involving the angle θ between two vectors:
v⋅w=∣v∣∣w∣cosθ📖
Equivalently, isolating cosθ:
cosθ=∣v∣∣w∣v1w1+v2w2+v3w3📖
The angle θ is measured between the heads of v and w:
Acute vs Obtuse Angles
AHL AI 3.13
If the scalar product of two vectors is negative, then
cosθ=∣u∣⋅∣v∣u⋅v<0
and thus θ must be an obtuse angle: 90°<θ≤180°.
But since vectors always form an acute AND an obtuse angle, we can find the acute angle by subtracting
180°−θ
whenever θ>90°.
Perpendicular vectors
AHL AI 3.13
Since v⋅w=∣v∣∣w∣cosθ, if two vectors are perpendicular then cos90°=0⇒ their scalar product is 0.
Properties of the scalar product
AHL AI 3.13
For any vectors u, v, and w, and scalar k:
u⋅v=v⋅u🚫
u⋅(v+w)=u⋅v+u⋅w🚫
(ku)⋅v=k(u⋅v)=u⋅(kv)🚫
u⋅u=∣u∣2🚫
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