Derivation and geometric interpretation of vector point multiplication formula

reproduced in http://blog.csdn.net/zsq306650083/article/details/8772128

1. Vector Point multiplication formula derivation and geometric interpretation

01. The vector point multiplication (dot product) is the sum of the products of each component, and the formula:

Write with a plus sign:

02. Geometrical Interpretation:

The result of a point multiplication is a scalar that is equal to the product of the vector size and the Cos value of the angle.

A B = |a| | b|cosθ

If both A and B are unit vectors, then the result of the point multiplication is the Cos value of the angle.

A B = cosθ

03. Derivation Process:

Suppose A and B are two-dimensional vectors, θ1 is the angle between a and X axis, θ2 is the angle between B and X axis, and the angle θ of vector A and B is equal to θ1-θ2.

A B = ax*bx + Ay*by

= (|a|*sinθ1) * (|b| * sinθ2) + (|a| * cosθ1) * (|b| * cosθ2)

= |a| | b| (sinθ1*sinθ2 + cosθ1*cosθ2)

=|a| | b| (cos (θ1-θ2))

= |a| | b|cosθ

2. Derivation of the point-multiplication exchange rate and distribution rate

01. Exchange Rate

Guan: This figure can be understood as: the geometric meaning of the point multiplication is that the projection of one edge to the other is multiplied by the length of the other side. December 11, 2016 add: A.B is a projection of B above A, B.A is a projection on the top of B.

02. Distribution Rate

Guan: This figure can be understood as: the geometric meaning of the point multiplication is that the projection of one edge to the other is multiplied by the length of the other side. December 2016 11 supplement: A-point multiply C can be understood as a projection of C on a.

Note: For more information see: <3d Math Primer for Graphics and game development second edition> click to open link