Planes And Axes

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In mathematics, a planes is the any flat, two-dimensional surface. A planes is the two dimensional analogues of a point (zero-dimensions), a line (one-dimension) and a space (three-dimensions). Planes can be arising as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry. X and Y axes, or X, Y, and Z axes, perpendicular line used in the Cartesian coordinate system.(Source.Wikipedia)
Examples for Area of Planes:

Example 1:

A wall in the forms of a rectangle has base 15m and height 10m.to find the area of the rectangle planes.

Solution:

Let b = 19 and h = 10.

Then the area of the rectangle = b × h = 19 × 10

= 190 sq. metres.

Example 2:

Find (a) area of a square whose side is 12 cm.

Solution:

Side of the square, a = 12 cm

Area of the square, A = a2

= a × a

= 12 × 12

= 144 sq.cm (or 144 cm2)

Example 3:

Find the area of rectangles whose length is 2m and breadth is 70 cm.

Solution:

Length ...
... of the rectangle, l = 2 m or 200 cm

Breadth of the rectangle, b = 70 cm

(a) Area of the rectangle, A = l × b

= 200 cm × 70 cm

∴ Area = 14000 sq.cm.

Example 4:

In an area right triangle the lengths of sides containing the right angles are 20 cm and 21 cm

Solution:

Let b = 20 cm and h = 21 cm

Area of the right triangle, A =1/2 bh

=1/2 × 20 cm × 21 cm

= 210 sq.cm
Examples for Axes:

Plot the points A (3, 0), B (0, √2), C (4, − 4), D (3, 3), E (−2.5, 1), F (−1, −3), G (−1, 0) and H (0, −4). Also specify the quadrant in which the each point lies.

Solution:

The points are plotted in the coordinate plane of the x and y axes. The point A lies on the positive side of the x–axes, the point B lies on the positive side of the y–axes, the point C lies within the IV quadrant, the point D lies in the I quadrant, the point E lies in the II quadrant, the point F lies in the III quadrant, the point G lies on the negative side of the x–axes and the point H lies on the negative side of the y–axes.

In three dimensions the graph of a linear equation has more freedom,and it can be a line or a plane. In this section we examine the equations of lines and planes and their graphs in 3–dimensional space, discuss how to determine their equations from information known about them, and look at ways to determine intersections and distances in three dimensions. Lines and planes are the simplest graphs in three dimensions, and they are useful for a variety of geometric and algebraic applications. They are also the building blocks we need to find tangent lines to curves and tangent planes to surfaces in three dimensions.
Lines in three Dimensions

Early in calculus we used the point-slope formula to write the equation of the line through a given point P = ( x0, y0 ) with a given slope m: y – y0 = m(x – x0)

Parametric Equation of a Line in Two Dimensions:

An equation of a line through the point P = ( x0, y0 ) and parallel to the vector A = 〈 a, b 〉 is given by the parametric equations

x = x(t) = x0 + a.t and

y = y(t) = y0 + b.t .

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Sample diagram for lines in two dimension:

line in two dimension

Notice that the coordinates of the point ( x0, y0 ) become the constant terms in the parametric equations,and the components of the vector 〈 a, b 〉 become the coefficients of the variable terms in the parametric equations. This same pattern is true for lines in three (and more) dimensions.

An equation of a line through the point P = ( x0, y0, z0 ) and parallel to the vector A = 〈 a, b, c 〉 is given by the parametric equations

x = x(t) = x0 + at,
y = y(t) = y0 + bt,
z = z(t) = z0 + ct .

Lines in three dimension sample diagram:

line in three dimension

Proof:

To show that P = ( x0, y0, z0 ) is on the line, put t = 0 and evaluate the three parametric equations:

x = x(0) = x0 + 0,

y = y(0) = y0 + 0, and

z = z(0) = z0 + 0.

To show that the line described by the parametric equations has the same direction as A, we pick another point Q on the line and show that the vector from P to Q is parallel to A.

Put t = 1, and let Q = ( x(1), y(1), z(1) ) = ( x0 + a, y0 + b, z0 + c ) .

Then the vector from P to Q is

V = 〈 (x0 + a) – x0, (y0 + b) – y0, (z0 + c) – z0 〉 = 〈 a, b, c 〉 = A

so the line described by the parametric equations is parallel to A .

Lines in Three Dimensions practice problems:

Find parametric equations for the line (a) through the point P = (3,0,2) and parallel to the vector A = 〈 1, 2, 0 〉 , and

(b) through the two points P = (3,0,2) and Q = (5, –1, 3).

Solution:

(a) By substituting in above equation we get,

x(t) = 3 + 1t,

y(t) = 0 + 2t,

z(t) = 2 + 0t .

(b) We can use the two points to get a direction for the line. The direction of the line is parallel to the vector from P to Q, 〈 2, –1,1 〉 , so

x(t) = 3 + 2t,

y(t) = 0 – 1t ,

z(t) = 2 + 1t .

Learn more on about Area of a Parallelogram Formula and its Examples. Between, if you have problem on these topics list of algebra 2 formulas, Please share your comments.