Vectors – The Dot Product Lecture 12 Mon, Sep 22, 2003.

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Presentation transcript:

Vectors – The Dot Product Lecture 12 Mon, Sep 22, 2003

The Dot Product The dot product of two vectors u = (u 1, …, u n ) v = (v 1, …, v n ) is u  v = u 1 v 1 + … + u n v n.

Properties of the Dot Product The dot product is a scalar. u  v = v  u. (tu)  v = u  (tv) = t(u  v). u  (v + w) = (u  v) + (u  w).

Dot Product and Angles The most important property of the dot product is that u  v = 0 if and only if u and v are perpendicular. More generally, u  v = |u||v|cos  where  is the angle between u and v.

Dot Product and Angles Therefore, u  v > 0 if and only if 0    < 90 . u  v = 0 if and only if  = 90 . u  v < 0 if and only if 90  <   180 . This is of the utmost importance in computer graphics.

Vectors and Scenes A polygonal face is not visible to the viewer if its normal vector makes more than a right angle with the vector to the viewer. Why? Is the converse true?

Vectors and Scenes A polygonal face is not lit by a light source if its normal vector makes more than a right angle with the vector to the light source. Why? Is the converse true?

Orthogonal Projections The orthogonal projection of u onto v is the vector [(u  v)/(v  v)]v Example: Project u = (1, 2) onto v = (3, 1). [(u  v)/(v  v)]v = (5/10)(3, 1) = (3/2, 1/2).

Reflections Let L be the “light” vector from the surface to the light source. Let N be the unit normal vector from the surface. Let R be the reflected vector.

Reflections L N R m e-e m R = -L + 2(L  N)N