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Equation of a Plane Parallel to a Given Plane

The equations...

Question

The equations of xy, yz and zx planes are _ , ___ and respectively.

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Solution

Equations of xy plane is given by z = 0
Since z is zero in xy plane
Similarly, equation of yz plane is given by x = 0 and equation of xz plane is given by y = 0
∴ Equation of xy, yz and zx planes are z = 0, x = 0 and y = 0 respectively.

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A_{x y}, A_{y z}, A_{z x}

be the area of projections of an area

A

x y, y z

z x

planes respectively, then considering

\to A

, a vector quantity whose direction in normal to surface

A

and its component along the coordinate axes be

A_{x}, A_{y}

A_{z}

\to A = A_{x} i + A_{y} j + A_{z} k

, therefore we can say

| A_{x} | = A_{z y}, ∣ ∣ A_{y} ∣ ∣ = A_{x z}, | A_{z} | = A_{x y}

and hence the area

∣ ∣ ∣ \to A ∣ ∣ ∣ = \sqrt{{A_{x}}^{2} + {A_{y}}^{2} + {A_{z}}^{2}} \Rightarrow A^{2} = {A_{x y}}^{2} + {A_{y z}}^{2} + {A_{z x}}^{2}

△

whose vertices are

(1, 2, 3), (2, 4, 1), (3, 4, 5)

respectively is

A_{x y}, A_{y z}, A_{z x}

be the area of projections of an area

A

x y, y z

z x

planes respectively, then considering

\to A

, a vector quantity whose direction in normal to surface

A

and its component along the coordinate axes be

A_{x}, A_{y}

A_{z}

\to A = A_{x} i + A_{y} j + A_{z} k

, therefore we can say

| A_{x} | = A_{z y}, ∣ ∣ A_{y} ∣ ∣ = A_{x z}, | A_{z} | = A_{x y}

and hence the area

∣ ∣ ∣ \to A ∣ ∣ ∣ = \sqrt{{A_{x}}^{2} + {A_{y}}^{2} + {A_{z}}^{2}} \Rightarrow A^{2} = {A_{x y}}^{2} + {A_{y z}}^{2} + {A_{z x}}^{2}

A plane makes intercepts

a, b, c

x, y, z

axes respectively, then twice the area of

△ A B C

including both its sides

(a, b, c

)

A_{x y}, A_{y z}, A_{z x}

be the area of projections of an area

A

x y, y z

z x

planes respectively, then considering

\to A

, a vector quantity whose direction in normal to surface

A

and its component along the coordinate axes be

A_{x}, A_{y}

A_{z}

\to A = A_{x} i + A_{y} j + A_{z} k

, therefore we can say

| A_{x} | = A_{z y}, ∣ ∣ A_{y} ∣ ∣ = A_{x z}, | A_{z} | = A_{x y}

and hence the area

∣ ∣ ∣ \to A ∣ ∣ ∣ = \sqrt{{A_{x}}^{2} + {A_{y}}^{2} + {A_{z}}^{2}} \Rightarrow A^{2} = {A_{x y}}^{2} + {A_{y z}}^{2} + {A_{z x}}^{2}

Through a point

P (h, k, l)

a plane is drawn at right angle to

O P

to meet the coordinate axes in

A, B, C

O P = p,

then the area of

△ A B C

Q. Let A,B,C be the feet of perpendiculars drawn from P(3,4,5) on XY,YZ,ZX planes respectively, then Coordinates of A,B and C will be .

Q. A variable plane passes through a fixed point

(3, 2, 1)

x, y

z

A, B

C

respectively. A plane is drawn parallel to

y z - p l a n e

A

, a second plane is drawn parallel

z x - p l a n e

B

and a third plane is drawn parallel to

x y - p l a n e

C

. Then the locus of the point of intersection of these three planes, is

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