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Dot Product of Two 3D Vectors (Components)

a⃗⋅b⃗=axbx+ayby+azbz\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y + a_z b_z

Worked example: (1, 2, 3)·(4, −5, 6) → 12 — press Try an example to run it live, then adjust anything.

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Dot Product of Two 3D Vectors (Components) explained

(ax, ay, az)(bx, by, bz)a·b

Adding a third axis costs the dot product exactly one more term. Its most-used property survives untouched: a·b = 0 means the two vectors meet at a right angle. That test underpins an enormous amount of working code — checking whether a surface normal faces a light, whether a ray runs parallel to a plane, whether two axes of a rotation matrix have drifted out of square. Example: (1, 2, 2)·(2, −2, 1) = 2 − 4 + 2 = 0, so those two are perpendicular even though neither lies along an axis and no picture would make it obvious.

Solve for one missing component and the equation is just a linear one in disguise: with a = (1, 2, 2), bx = 2, by = −2 and a required dot product of 0, the remaining component must satisfy 2 − 4 + 2bz = 0, giving bz = 1. The paired coefficient has to be non-zero, though — if az = 0 then bz never enters the sum, and no dot product can reveal it.

Dot Product of Two 3D Vectors (Components) formula

a⃗⋅b⃗=axbx+ayby+azbz\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y + a_z b_z
Where
  • a⃗⋅b⃗\vec{a}\cdot\vec{b}= Dot product
  • axa_x= x-component of a
  • aya_y= y-component of a
  • aza_z= z-component of a
  • bxb_x= x-component of b
  • byb_y= y-component of b
  • bzb_z= z-component of b

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