M408M Learning Module Pages
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Chapter 10: Parametric Equations
and Polar Coordinates
Chapter 12: Vectors and the Geometry of Space
Learning module LM 12.1:
3dimensional rectangular coordinates:
Learning module LM 12.2: Vectors:
Learning module LM 12.3: Dot products:
Learning module LM 12.4: Cross products:
Learning module LM 12.5: Equations of Lines and Planes:
Equations of a line
Equations of planes
Finding the normal to a plane
Distances to lines and planes
Learning module LM 12.6: Surfaces:
Chapter 13: Vector Functions
Chapter 14: Partial Derivatives
Chapter 15: Multiple Integrals


Distances to lines and planes
Distances to Lines and Planes
Using dot products, we can compute
distances between geometric objects.
 Let ${\bf n}$ be the normal
vector for a plane through the point $Q$. The distance from another
point $P$ to this plane is $\displaystyle{\frac{\overrightarrow{Q\,P}
\cdot {\bf n}}{\ {\bf n}\}}$.
 The distance from the plane
$Ax+By+Cz=D$ to the point $P(x_1,y_1,z_1)$ is
$\displaystyle{\frac{Ax_1 + By_1 + C z_1
D}{\sqrt{A^2+B^2+C^2}}}$.
 The distance from the plane $Ax + By + Cz = D$ to the parallel
plane $Ax + By + Cz = D'$ is
$\displaystyle{\frac{DD'}{\sqrt{A^2+B^2+C^2}}}$
 The distance from a line with vector equation ${\bf r}(t) = {\bf
b} + t {\bf v}$ to a point $P$ with position vector ${\bf u}$ is
$\displaystyle{\frac {\ {\bf v} \times ({\bf u}{\bf b})\}{\ \bf v
\}}$.
 The smaller angle between the plane $A_1 x + B_1 y + C_1 z = D_1$
and the plane $A_2 x + B_2 y + C_2 z = D_2$ is either the angle
between the normal vectors ${\bf n_1} = \langle A_1, B_1, C_1 \rangle$
and ${\bf n_2} = \langle A_2, B_2, C_2 \rangle$, or is $\pi$ minus
that angle, whichever is less. The cosine of the angle between the
planes is $\displaystyle{\frac{{\bf n_1}\cdot {\bf n_2}}{\{\bf
n_1}\ \{\bf n_2}\}}$.
