Linear Algebra

Dot Product

Drop b perpendicularly onto a and you get a shadow. The dot product is the length of a times that shadow, and nothing else. Drag the arrows and watch the number follow the picture.

abshadow of b on a

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Dot product—

—

opposed—right angles0same direction—

View

Scenario

Vector a

x 2.00

y 0.60

z 0.00

Vector b

x 0.80

y 1.80

z 0.00

Angle between them, swinging b —

Spin the scene

Values & calculations

length of a—

length of b—

shadow of b on a—

angle between—

What to observe

At right angles the shadow of b on a vanishes and a · b = 0. Nothing of one goes along the other.
Swing b past 90°: the foot of the perpendicular lands behind the origin, the shadow counts as negative, and so does the product.
Give a length 1 and a · b is the shadow, with no scaling left over: "how much of b lies along this direction".
In 3D one more term a_zb_z joins the component formula, and the picture is the same shadow.

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