Dot Product (Projection)
Given two vectors a⇀ and b⇀, their dot product is the scalar quantity
a⇀ ⋅ b ⇀= a⋅|b|⋅cosθ
where θ is the angle between a⇀ and b⇀.
The dot product can also be expressed in terms of the components of a⇀ and b⇀ as follows:
a⇀⋅b ⇀= a1⋅b1 + a2 ⋅b2 +a3⋅ b3
The unit vector in the direction of a⇀ is given by
The vector projection of a⇀ on b⇀ is the orthogonal projection of a⇀ onto the line in the direction of b⇀:
a⋅cosθ⋅b^ = a⇀⋅b^⋅b^ =a⇀⋅b ⇀⋅b ⇀b2= ax⋅bx + ay ⋅by + az⋅bzbx2+ by2 +by2 ⋅ b ⇀
The scalar projection of a⇀ on b⇀ is the length of the associated vector projection.
a⋅cosθ = a⇀⋅b^ =a⇀⋅b ⇀|b| = ax⋅bx + ay ⋅by + az⋅bzbx2+ by2 +by2
Click and or drag on the graph to change the two vectors. See how they affect the scalar and vector projections.
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