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Vectors – The Dot Product

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 .

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Vectors – The Dot Product

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

  2. The Dot Product • The dot product of two vectors u = (u1, …, un) v = (v1, …, vn) is u v = u1v1 + … + unvn.

  3. 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).

  4. 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.

  5. 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.

  6. 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?

  7. 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?

  8. 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).

  9. 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.

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

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