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# Dot product.

Definition 1.6   Let and be two vectors in . The dot product, or scalar product of and is the real number

Proposition 1.7

1.
, .
2.
.
3.
,
4.
, .
5.
, .
6.
, ,

Proposition 1.8   Two vectors and are orthogonal if, and only if, .

Example 1.9   If and , then . Thus .

Definition 1.10   Let be a plane whose cartesian equation is ax+by+cz+d=0. The vector is called a normal vector for .

Example 1.11   The vector is normal to the plane whose equation is x + 2y -3z +5 =0.

Proposition 1.12   Let and be two planes with respective normal vectors and . The two planes and are orthogonal if, and only if, the vectors and are orthogonal.

Example 1.13   Let and . These planes have respective normal vectors and . We have: , thus .

Proposition 1.14   A line and a plane are perpendicular if, and only if, a direction vector for is a normal vector for .

Example 1.15   Let be the line given by the parametric equations

and let be the plane whose cartesian equation is 2x+4y-3z-1=0. Then and are perpendicular.

Example 1.16   If is defined by the equation 3x-y+5z-6=0, then the line through the point A(5,-2,7) and perpendicular to has the following parametric equations:

Next: Cross product. Up: Analytic geometry in the Previous: Planes and lines.
Noah Dana-Picard
2001-05-30