Projection of the Join of Two Points

In 3D geometry, the projection of the join of two points refers to the image of the line segment connecting those two points onto a particular plane or line. This concept is important in various fields such as computer graphics, engineering, and mathematics. Understanding how to calculate and interpret this projection is crucial for solving problems related to 3D space.

Understanding Projections

Before we delve into the specifics of projecting the join of two points, let's understand what a projection is. A projection is a mapping of a point, line, or figure from a higher-dimensional space onto a lower-dimensional subspace, such that the image retains certain properties of the original.

Types of Projections

• Orthogonal Projection: The projection is made perpendicular to the plane or line onto which the figure is being projected.
• Oblique Projection: The projection is made at an angle other than 90 degrees to the plane or line.

Projection onto a Plane

When projecting the join of two points onto a plane, we are looking for the intersection of the line segment with the plane or its extension.

Formula for Projection onto a Plane

Let's consider two points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ in 3D space. The vector $\vec{AB}$ representing the join of $A$ and $B$ is given by:

$$ \vec{AB} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle $$

If we want to project this vector onto a plane with a normal vector $\vec{n} = \langle a, b, c \rangle$, we can use the following formula:

$$ \text{Projection of } \vec{AB} \text{ onto plane } = \vec{AB} - \frac{\vec{AB} \cdot \vec{n}}{|\vec{n}|^2} \vec{n} $$

Where $\cdot$ denotes the dot product and $|\vec{n}|$ is the magnitude of the normal vector.

Example of Projection onto a Plane

Let's project the join of points $A(1, 2, 3)$ and $B(4, 5, 6)$ onto the plane with the normal vector $\vec{n} = \langle 1, 1, 1 \rangle$.

1. Calculate $\vec{AB}$:

$$ \vec{AB} = \langle 4 - 1, 5 - 2, 6 - 3 \rangle = \langle 3, 3, 3 \rangle $$

1. Calculate the dot product $\vec{AB} \cdot \vec{n}$:

$$ \vec{AB} \cdot \vec{n} = 3 \cdot 1 + 3 \cdot 1 + 3 \cdot 1 = 9 $$

1. Calculate the magnitude of $\vec{n}$:

$$ |\vec{n}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} $$

1. Calculate the projection:

$$ \text{Projection of } \vec{AB} \text{ onto plane } = \langle 3, 3, 3 \rangle - \frac{9}{3} \langle 1, 1, 1 \rangle = \langle 0, 0, 0 \rangle $$

In this case, the projection is a point because the line segment is parallel to the plane's normal vector.

Projection onto a Line

When projecting the join of two points onto a line, we are looking for the shadow that the line segment would cast on the line if light were shining perpendicular to it.

Formula for Projection onto a Line

To project $\vec{AB}$ onto a line with direction vector $\vec{d} = \langle d_1, d_2, d_3 \rangle$, we use the formula:

$$ \text{Projection of } \vec{AB} \text{ onto line } = \frac{\vec{AB} \cdot \vec{d}}{|\vec{d}|^2} \vec{d} $$

Example of Projection onto a Line

Let's project the join of points $A(1, 2, 3)$ and $B(4, 5, 6)$ onto the line with direction vector $\vec{d} = \langle 1, 0, 0 \rangle$.

1. Calculate $\vec{AB}$:

$$ \vec{AB} = \langle 4 - 1, 5 - 2, 6 - 3 \rangle = \langle 3, 3, 3 \rangle $$

1. Calculate the dot product $\vec{AB} \cdot \vec{d}$:

$$ \vec{AB} \cdot \vec{d} = 3 \cdot 1 + 3 \cdot 0 + 3 \cdot 0 = 3 $$

1. Calculate the magnitude of $\vec{d}$:

$$ |\vec{d}| = \sqrt{1^2 + 0^2 + 0^2} = 1 $$

1. Calculate the projection:

$$ \text{Projection of } \vec{AB} \text{ onto line } = \frac{3}{1} \langle 1, 0, 0 \rangle = \langle 3, 0, 0 \rangle $$

The projection of the join of $A$ and $B$ onto the line is the vector $\langle 3, 0, 0 \rangle$.

Table of Differences

Aspect	Projection onto a Plane	Projection onto a Line
Dimensionality	2D (plane)	1D (line)
Direction	Perpendicular to normal	Along direction vector
Formula	$\vec{AB} - \frac{\vec{AB} \cdot \vec{n}}{\	\vec{n}\
Example Projection	$\langle 0, 0, 0 \rangle$ (point)	$\langle 3, 0, 0 \rangle$ (vector)
Application	Shadow on a plane	Shadow on a line

Understanding the projection of the join of two points is essential for visualizing and solving problems in 3D space. By using the formulas provided and following the examples, one can effectively project line segments onto planes and lines for various applications.