By Jazar R.N.

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273) Example 49 Plane through Three Points Every three points indicate a plane. Assume that (x1 , y1 , z1 ), (x2 , y2 , z2 ), and (x3 , y3 , z3 ) are the coordinates of three points P1 , P2 , and P3 . 277) The determinant of the equations must be zero, which determines the equation of the plane. 282) or is normal to the plane and may be used to represent the plane. Example 51 Quadratic Surfaces A quadratic relation between x, y, z is called the quadratic form and is an equation containing only terms of degree 0, 1, and 2 in the variables x, y, z.

Two vectors are equal if they are comparable and are the same type and have the same characteristics. Two vectors are equivalent if they are comparable and the same type and can be substituted with each other. In summary, any physical quantity that can be represented by a directed section of a line with a start and an end point is a vector quantity. A vector may have five characteristics: length, axis, end point, direction, and physical quantity. The length and direction are necessary. There are seven types of vectors: vecpoint, vecline, vecface, vecfree, vecpoline, vecpoface, and vecporee.

Length and direction are necessary to have a vector; however, a vector may have five characteristics: 1. Length. The length of section OP corresponds to the magnitude of the physical quantity that the vector is representing. 2. Axis. A straight line that indicates the line on which the vector is. The vector axis is also called the line of action. 3. End point. A start or an end point indicates the point at which the vector is applied. Such a point is called the affecting point. 4. Direction. The direction indicates at what direction on the axis the vector is pointing.

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