# Find Point D on Plane for 4 Unit Cube Pyramid

• MHB
• jaychay
In summary, point $d$ is located at (-12, 12, 12) and the height of the pyramid is $12\sqrt{2}$ units. The distance from point $d$ to the plane is calculated using the formula $d = \dfrac{|Ax_1+By_1+Cz_1+D|}{\sqrt{A^2+B^2+C^2}}$ and the height is calculated using the formula $V = \dfrac{Bh}{3}$.
jaychay
Find point d on the line of r(t)=(0,0,0)+(−1,1,1)t which make the triangular pyramid abcd has the volume of 4 unit cube when a(0,0,0),b(1,0,1),c(0,1,0) are the points on the plane of −x+z=0.

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The area $B$ of the base rectangular triangle $\Delta abc$ is $B=\frac 12\cdot ab\cdot ac = \frac 12 \sqrt 2\cdot 1$.
The volume of the pyramid is $V=\frac 13 Bh$, where $B$ is the area of the base, and $h$ is the height perpendicular to the base.
The normal vector $\vec n$ of the plane can be deduced from its equation $-x+z=0$, meaning it is $\vec n=(-1,0,1)$.
The height $h$ of the pyramid is the projection of the vector $\overrightarrow{ad}$ onto the normal vector $\vec n$.
The formula for that projection is $h=\frac{\overrightarrow{\mathstrut ad} \cdot \overrightarrow{\mathstrut n}}{\|\vec n\|}$.

So we have:
$$\begin{cases}B=\frac 12\sqrt 2 \\ V=\frac 13 Bh = 4 \\ \vec n = (-1,0,1) \\ h=\frac{\overrightarrow{\mathstrut ad} \cdot \overrightarrow{\mathstrut n}}{\|\vec n\|} = \frac{(-1,1,1)t \cdot (-1,0,1)}{\|(-1,0,1)\|} = \frac{2}{\sqrt 2}t=t\sqrt 2 \end{cases} \implies V = \frac 13 \cdot \frac 12\sqrt 2 \cdot t\sqrt 2 = 4 \implies t = 12$$
So point $d$ is $(0,0,0)+(-1,1,1)12=(-12,12,12)$.

Thank you very much !

I learned (long ago) this formula for the distance from a point, $(x_1,y_1,z_1)$ to a plane $Ax+By+Cz+D = 0$

$d = \dfrac{|Ax_1+By_1+Cz_1+D|}{\sqrt{A^2+B^2+C^2}}$

$V = \dfrac{Bh}{3} = \dfrac{1}{\sqrt{2}} \cdot \dfrac{h}{3} = 4 \implies h = 12 \sqrt{2}$

$12\sqrt{2} = \dfrac{|-1(-t) + 0(t) + 1(t) + 0|}{\sqrt{(-1)^2 + 0^2 + 1^2}}$

$12\sqrt{2} = \dfrac{2t}{\sqrt{2}} \implies t = 12$

## 1. How do you find the coordinates for point D on a plane for a 4 unit cube pyramid?

To find the coordinates for point D, you will need to use a coordinate plane and the dimensions of the cube pyramid. First, plot the coordinates for points A, B, and C on the plane. Then, use the Pythagorean theorem to find the length of the diagonal edge connecting point A to point C. This length will be equal to the length of the diagonal edge connecting point B to point D. Use this information to find the coordinates for point D.

## 2. What is the purpose of finding point D on a plane for a 4 unit cube pyramid?

The purpose of finding point D is to accurately represent the location and dimensions of the 4 unit cube pyramid on a coordinate plane. This can be useful in various mathematical and scientific applications, such as calculating volume or surface area.

## 3. Can point D be located anywhere on the plane for a 4 unit cube pyramid?

No, point D must be located in a specific location on the plane in order for the cube pyramid to maintain its shape and dimensions. The coordinates for point D will depend on the coordinates of points A, B, and C, as well as the dimensions of the cube pyramid.

## 4. Is it possible to find point D without using a coordinate plane?

Yes, it is possible to find point D without using a coordinate plane. This can be done by using other mathematical methods, such as using the coordinates of points A, B, and C, and the dimensions of the cube pyramid to calculate the coordinates for point D.

## 5. How can finding point D on a plane for a 4 unit cube pyramid be applied in real life?

Finding point D on a plane for a 4 unit cube pyramid can be applied in real life in various engineering and construction projects. For example, it can be used to accurately represent the dimensions and location of a building or structure, which can be useful in the planning and design process.

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