3D Coordinate Geometry - Distance
In 3D geometry, the distance between two objects is the length of the shortest line segment connecting them; this is analogous to the two-dimensional definition.
Contents
Distance between 2 Points
In three-dimensional space, points are represented by their positions along the \(x\)-, \(y\)-, and \(z\)-axes, which are each perpendicular to one another; this is analogous to the 2d coordinate geometry interpretation in which each point is represented by only two coordinates (along the \(x\)- and \(y\)-axes).
dk
In the figure above, the goal is to find the distance from the point \(\left(x_{1},y_{1},z_{1}\right)\) to the point \(\left(x_{2},y_{2},z_{2}\right).\) From the distance formula in two dimensions, the length of the the yellow line is
\[\sqrt { { ({ x }_{ 2 }-{ x }_{ 1 }) }^{ 2 }+{ ({ y }_{ 2 }{ -y }_{ 1 }) }^{ 2 } }. \]
Let \(d\) be the distance from the point \(\left(x_{1},y_{1},z_{1}\right)\) to \(\left(x_{2},y_{2},z_{2}\right)\) (the red line, and the desired distance). Then, using the Pythagorean theorem,
\[\begin{align} {d }^{ 2 }&= \left(\sqrt { { ({ x }_{ 2 }-{ x }_{ 1 }) }^{ 2 }+({ { y }_{ 2 }-{ y }_{ 1 }) }^{ 2 } } \right)^{ 2 }+{ ({ z }_{ 2 }-{ z }_{ 1 }) }^{ 2 }\\ \Rightarrow d&=\sqrt { { ({ x }_{ 2 }-{ x }_{ 1 }) }^{ 2 }+({ { y }_{ 2 }-{ y }_{ 1 }) }^{ 2 }+{ ({ z }_{ 2 }-{ z }_{ 1 }) }^{ 2 } }. \end{align}\]
The above equation is the general form of the distance formula in 3D space. A special case is when the initial point is at the origin, which reduces the distance formula to the form
\[d=\sqrt { { { x } }^{ 2 }+{ { y } }^{ 2 }+{ { z } }^{ 2 } },\]
where \((x,y,z)\) is the terminal point. This equation extends the distance formula to 3D space.
Find the distance between the points \((2,-5,7)\) and \((3,4,5).\)
From the distance formula, we have
\[\begin{align} d&=\sqrt { { (3-2) }^{ 2 }+\big({ 4-(-5)\big) }^{ 2 }+{ (5-7) }^{ 2 } } \\ &=\sqrt { 1+81+4 } \\ &=\sqrt { 86 }.\ _\square \end{align}\]
How far is the point \(P=(3,4,5)\) from the origin?
Since the second point is the origin, or \((0,0,0)\), the distance is
\[d=\sqrt{3^2+4^2+5^2}=5\sqrt{2}.\ _\square\]
If the distance between the two points \((2,0,3)\) and \((-4,a,-1)\) is 8, what is the value of \(a?\)
From the distance formula,
\[\begin{align*} d&=\sqrt{\big(2-(-4)\big)^2+(0-a)^2+\big(3-(-1)\big)^2}\\ &=\sqrt{36+a^2+16}\\ &=\sqrt{52+a^2}\\ &=8. \end{align*} \]
Therefore,
\[\begin{align*} 52+a^2&=8^2\\&=64\\ a&=\pm2\sqrt{3}.\ _\square \end{align*}\]
Distance between a Point and a Plane
Determining the distance between a point and a plane follows a similar strategy to determining the distance between a point and a line. Consider a plane defined by the equation
\[ax + by + cz + d = 0\]
and a point \((x_0, y_0, z_0)\) in space. Then the normal vector to the plane is
\[\mathbf{v} = \begin{pmatrix}a\\b\\c\end{pmatrix}\]
and the vector from an arbitrary point on the plane \((x,y,z)\) to the point is
\[\mathbf{w} = \begin{pmatrix}x_0-x\\y_0-y\\z_0-z\end{pmatrix}.\]
The distance from the point to the plane is the projection from \(\mathbf{w}\) onto \(\mathbf{v}\), or
\[ \begin{align*} D &= |\text{proj}_{\mathbf{v}}\mathbf{w}| \\ &= \frac{|\mathbf{v} \cdot \mathbf{w}|}{|\mathbf{v}|} \\ &= \frac{|a(x_0-x)+b(y_0-y)+c(z_0-z)|}{\sqrt{a^2+b^2+c^2}} \\ &= \frac{|ax_0+by_0+cz_0-(ax+by+cz)|}{\sqrt{a^2+b^2+c^2}} \\ &= \frac{|ax_0+by_0+cz_0+d|}{\sqrt{a^2+b^2+c^2}}. \end{align*} \]
A planes passes through the point \((1,-2,3)\) and is parallel to the plane \(2x-2y+z=0\). The distance of the point \((-1,2,0)\) from the plane is \(\text{__________}.\)
Distance between 2 Skew Lines
The strategy behind determining the distance between 2 skew lines is to find two parallel planes passing through each line; this is because the distance between two planes is easy to calculate using vector projection. Furthermore, the normal vector to these 2 planes can be calculated using the cross product of the vectors representing the direction of the two lines.
In particular, suppose the two lines travel in the directions
\[\begin{align} \mathbf{v}&=\begin{pmatrix}x_1&y_1&z_1\end{pmatrix}\\ \mathbf{w}&=\begin{pmatrix}x_2&y_2&z_2\end{pmatrix}, \end{align}\]
then the normal to the planes can be calculated as
\[\mathbf{n} = \mathbf{v} \times \mathbf{w} = \text{det}\begin{pmatrix}\hat{i}&\hat{j}&\hat{k}\\x_1&y_1&z_1\\x_2&y_2&z_2\end{pmatrix}\]
and the distance between the two planes--which is the same as the distance between the two lines--can be calculated by projecting \(\mathbf{n}\) onto \(PQ\), where \(P\) and \(Q\) are points on the first and second lines, respectively.
Find the distance between the lines
\[\frac{x+2}{2}=\frac{y-1}{3}=\frac{z}{1}\quad \text{and}\quad \frac{x-3}{-1}=\frac{y}{1}=\frac{z+1}{2}.\]
Following the above strategy, the first line travels in the direction
\[\mathbf{v}=\begin{pmatrix}2&3&1\end{pmatrix}\]
and the second in the direction
\[\mathbf{w}=\begin{pmatrix}-1&1&2\end{pmatrix}.\]
Then
\[\mathbf{n} = \mathbf{v} \times \mathbf{w} = \text{det}\begin{pmatrix}\hat{i}&\hat{j}&\hat{k}\\2&3&1\\-1&1&2\end{pmatrix}=\begin{pmatrix}5&-5&5\end{pmatrix}.\]
Since \((-2, 1, 0)\) and \((3, 0, -1)\) are points on the two lines, respectively, the vector \(PQ\) is \(\begin{pmatrix}5&-1&-1\end{pmatrix}\). Then
\[D=\big|\text{proj}_{\mathbf{n}}PQ\big|=\frac{\big|\mathbf{n} \cdot PQ\big|}{|\mathbf{n}|}=\frac{25}{5\sqrt{3}}=\frac{5\sqrt{3}}{3}.\ _\square\]
Find the shortest distance between the following two lines \({g}_{1}\) and \({g}_{2}:\)
\[ \begin{align} {g}_{1} &: \frac{x-3}{2} = \frac{y+1}{-2} = z-2 \\ {g}_{2} &: x = \frac{y}{2} = -z+4. \end{align}\]