1: Vectors and Geometry in Two and Three Dimensions
- Page ID
- 89202
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Before we get started doing calculus in two and three dimensions we need to brush up on some basic geometry, that we will use a lot. We are already familiar with the Cartesian plane 1, but we'll start from the beginning.
- René Descartes (1596–1650) was a French scientist and philosopher, who lived in the Dutch Republic for roughly twenty years after serving in the (mercenary) Dutch States Army. He is viewed as the father of analytic geometry, which uses numbers to study geometry.
- 1.1: Points
- Each point in two dimensions may be labeled by two coordinates \((x,y)\) which specify the position of the point in some units with respect to some axes as in the figure below.
- 1.2: Vectors
- In many of our applications in 2d and 3d, we will encounter quantities that have both a magnitude (like a distance) and also a direction. Such quantities are called vectors. That is, a vector is a quantity which has both a direction and a magnitude, like a velocity.
- 1.3: Equations of Lines in 2d
- A line in two dimensions can be specified by giving one point \((x_0,y_0)\) on the line and one vector \(\textbf{d}=\left \langle d_x,d_y \right \rangle \) whose direction is parallel to the line.
- 1.4: Equations of Planes in 3d
- Specifying one point \((x_0,y_0,z_0)\) on a plane and a vector \(\textbf{d}\) parallel to the plane does not uniquely determine the plane, because it is free to rotate about \(\textbf{d}\text{.}\)\vd\text{.}
- 1.5: Equations of Lines in 3d
- Just as in two dimensions, a line in three dimensions can be specified by giving one point \((x_0,y_0,z_0)\) on the line and one vector \(\textbf{d}=\left \langle d_x,d_y,d_z \right \rangle \) whose direction is parallel to that of the line.
- 1.6: Curves and their Tangent Vectors
- The right hand side of the parametric equation \((x,y,z)=(1,1,0)+t\left \langle 1,2,-2 \right \rangle\) that we just saw in Warning 1.5.3 is a vector-valued function of the one real variable \(t\text{.}\)
- 1.7: Sketching Surfaces in 3d
- In practice students taking multivariable calculus regularly have great difficulty visualising surfaces in three dimensions, despite the fact that we all live in three dimensions. We'll now develop some technique to help us sketch surfaces in three dimensions.
- 1.8: Cylinders
- There are some classes of relatively simple, but commonly occurring, surfaces that are given their own names. One such class is cylindrical surfaces. You are probably used to thinking of a cylinder as being something that looks like \(x^2+y^2=1\text{.}\)
- 1.9: Quadric Surfaces
- Another named class of relatively simple, but commonly occurring, surfaces is the quadric surfaces.