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Math Formulas: Planes in three dimensions Plane forms Point direction form: 1. a(x x 1)+b(y y 1)+c(z z 1) = 0 where P(x 1;y 1;z 1) lies in the plane, and the direction (a;b;c) is normal to the plane. General form: 2. Ax+By+Cz+D= 0 where direction (A;B;C) is normal to the plane. Intercept form: 3. x a + y b + z c = 1
Example 1: Find parametric equations for the lines through the point P = (1,2) that are (a) parallel to the vector A = 〈 3, 5 〉 , and (b) parallel to the vector B = 〈 6, 10 〉 . Then graph the two lines.
Lines are parallel if they are in the same plane and they never intersect. Lines f and g, at right, are parallel. Lines are perpendicular if they intersect at a 90⁰ angle. A pair of perpendicular lines is always in the same plane. Lines f and e, at right, are perpendicular. Lines g and e are also perpendicular.
Here are some examples of pairs of lines in a coordinate plane. a. 2 x + y = 2 These lines are not parallel b. 2 x + y = 2 These lines are coincident x − y = 4 or perpendicular.
To write the equation of a line in 3D space, we need a point on the line and a parallel vector to the line. Example 1: Find the vector, parametric, and symmetric equations for the line through the point (1, 0, -3) and parallel to the vector 2 i - 4 j + 5 k.
In a plane, 2 lines that are perpendicular to a common line are parallel. Two lines that are perpendicular to a common line must be parallel. A triangle is always a planar figure.
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Find an equation of the plane through the points (0; 1; 1); (1; 0; 1) and (1; 1; 0): 4. Find an equation of the line through the point ( 2; 4; 10) and parallel to the vector h3; 1; 8i: Check if (4; 6; 6) and (1; 4; 4) are on the line. 5. Find an equation of the line through the point (1; 0; 6) and perpendicular to the plane x + 3y +. 6.
