12.2 Three-Dimensional Space: Surfaces, Vectors, and Curves (Wk 1)

State the distance formula in $\mathbb{R}^3$ (Theorem 1) and its special case from the origin.

• If $P=(a_1,b_1,c_1)$ and $Q=(a_2,b_2,c_2)$, the distance between $P$ and $Q$ is
  $|P-Q|=\sqrt{(a_2-a_1)^2+(b_2-b_1)^2+(c_2-c_1)^2}$

• Special case: $Q=O=(0,0,0)$ and $P=(a,b,c)$ gives
  $|OP|=\sqrt{a^2+b^2+c^2}$

Give the equations of planes parallel to the coordinate planes.

• xy-plane: $\{(x,y,0):x,y\in\mathbb{R}\}$, equation $z=0$.

• Plane parallel to the xy-plane: $\{(x,y,c):x,y\in\mathbb{R}\}$, equation $z=c$.

• Similarly:

  • $x=a$ is a plane parallel to the yz-plane.

  • $y=b$ is a plane parallel to the xz-plane.

Give the set description and equation of a sphere.

• A sphere centered at $Q=(a,b,c)$ with radius $R$:
  $\{(x,y,z):(x-a)^2+(y-b)^2+(z-c)^2=R^2\}$

• Equation: $(x-a)^2+(y-b)^2+(z-c)^2=R^2$.

Give the equation of a right cylinder in $\mathbb{R}^3$, and compare it with a circle in $\mathbb{R}^2$.

• A right cylinder in $\mathbb{R}^3$ has radius $R$ and central axis the vertical line through $(a,b,0)$.

• $R$ is the distance between the point $(x,y,z)$ and that vertical line (the closest point on the line is $(a,b,z)$).

• Set: $\{(x,y,z):(x-a)^2+(y-b)^2=R^2\}$.

• Equation (for 3D space): $(x-a)^2+(y-b)^2=R^2$.

• Compare: a circle in $\mathbb{R}^2$ with radius $R$ and center $(a,b)$ is $\{(x,y):(x-a)^2+(y-b)^2=R^2\}$. It is the same equation but for 2D space, with only two coordinates.

How do vectors in 3-space work (components, length, addition, scalar multiplication)?

• Geometrically it is almost the same as 2D, but pictures are harder to draw.

• Algebraically there are just 3 coordinates/components.

• For $P=(a_1,b_1,c_1)$ and $Q=(a_2,b_2,c_2)$:

  • Vector: $\overrightarrow{PQ}=\langle a_2-a_1,\,b_2-b_1,\,c_2-c_1\rangle$ (these are the components).

  • Length: $\|\overrightarrow{PQ}\|=\sqrt{(a_2-a_1)^2+(b_2-b_1)^2+(c_2-c_1)^2}$.

• For $\vec v=\langle v_1,v_2,v_3\rangle$, $\vec w=\langle w_1,w_2,w_3\rangle$ and $\lambda\in\mathbb{R}$:

  • Addition: $\vec v+\vec w=\langle v_1+w_1,\,v_2+w_2,\,v_3+w_3\rangle$.

  • Scalar multiplication: $\lambda\vec v=\langle\lambda v_1,\lambda v_2,\lambda v_3\rangle$.

Define parallel vectors in 3-space and the standard basis $\vec i,\vec j,\vec k$. How do you write any vector in terms of it?

• Parallel: non-zero $\vec v,\vec w$ are parallel if there exists $\lambda\in\mathbb{R}$ with $\vec v=\lambda\vec w$.

• Standard basis in $\mathbb{R}^3$: $\vec i=\langle1,0,0\rangle$, $\vec j=\langle0,1,0\rangle$, $\vec k=\langle0,0,1\rangle$.

• Any vector in $\mathbb{R}^3$ is a linear combination of $\vec i,\vec j,\vec k$:
  $\langle a,b,c\rangle=\langle a,0,0\rangle+\langle0,b,0\rangle+\langle0,0,c\rangle=a\vec i+b\vec j+c\vec k$

How do you describe a general curve in 3D space with a parametric equation?

• For a point $P$ on the curve, write $\vec r=\overrightarrow{OP}$.

• Parametric equation: $\vec r=\vec r(t)=\langle x(t),y(t),z(t)\rangle$, or equivalently the system
  $x=x(t),\qquad y=y(t),\qquad z=z(t)$

• The notes’ picture marks points on the curve at $t=0$ and $t=1$ (position vectors $\vec r(0)$ and $\vec r(1)$ from $O$).

Give the equation of a line through a point $P=(x_0,y_0,z_0)$ with direction $\vec v=\langle a,b,c\rangle$. Show how it is derived.

• Derivation: for a point $Q=(x,y,z)$ on the line, the vector from the base point to $Q$ is parallel to $\vec v$. So there is $t\in\mathbb{R}$ with (vector from base point to $Q$) $=t\vec v$. (The notes write this vector as $\overrightarrow{PP_0}$, which is presumably the vector between the base point and $Q$.) Thus
  $\langle x-x_0,\,y-y_0,\,z-z_0\rangle=t\langle a,b,c\rangle$

• Vector form: $\langle x,y,z\rangle=\langle x_0,y_0,z_0\rangle+t\langle a,b,c\rangle$.

• Writing $\vec r=\overrightarrow{OQ}$: $\vec r(t)=\overrightarrow{OP}+t\vec v$.

• Parametric (scalar) form:
  $x=x_0+at,\qquad y=y_0+bt,\qquad z=z_0+ct$

Give the equation of the line through two points $P=(a_1,b_1,c_1)$ and $Q=(a_2,b_2,c_2)$.

• Take the direction $\vec v=\overrightarrow{PQ}=\langle a_2-a_1,\,b_2-b_1,\,c_2-c_1\rangle$.

• Equation of the line: $\vec r(t)=\overrightarrow{OP}+t\,\overrightarrow{PQ}$.

• Component form: $\vec r(t)=\langle a_1,b_1,c_1\rangle+t\langle a_2-a_1,\,b_2-b_1,\,c_2-c_1\rangle$.

• Parametric form:
  $x=a_1+(a_2-a_1)t,\qquad y=b_1+(b_2-b_1)t,\qquad z=c_1+(c_2-c_1)t$