Section 2.2 - Vectors in three dimensions

Section Objectives

1. Distinguish 3-dimensional space from 2-dimensional space.
2. Work with the rectangular coordinate system in 3-space.
3. Perform and describe basic operations on vectors in 3-space.
4. Compute the magnitude of a vector in 3-space.

Introduction

To move beyond 2-space into 3-dimensional space, we introduce the $z$-axis into our standard rectangular coordinate system. The $x$-, $y$-, and $z$-axes are mutually perpendicular, sharing the common point at the origin. Points in space are identified with three ordered triplets, $(x,y,z)$.

The arrangement of the axes satisfies the right-hand rule: with your right hand, point your fingers in the direction of the positive $x$-s, curl your fingers toward the positive $y$-axis, and your thumb will point in the direction of the positive $z$-axis. However you envision (or sketch) your axes, make sure they follow the right-hand rule!

Examples

• Distance formula: The distance from $(x_1,y_1,z_1)$ to $(x_2,y_2,z_2)$ is given by $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$. Find the distance from $P\,(4,-3,-2)$ to $Q\,(9,4,-5)$.

• Midpoint formula: The midpoint of the segment connecting $(x_1,y_1,z_1)$ to $(x_2,y_2,z_2)$ is $\displaystyle \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2} \right)$. Find the midpoint of the segment $PQ$, where $P$ and $Q$ are the points above.

• The sphere with center $(a,b,c)$ and radius $r$ is described in standard form by the equation $(x-a)^2+(y-b)^2+(z-c)^2=r^2$. Find the center and radius of the sphere described by $x^2+y^2+z^2+2x-6y=6$.

• Describe the surface in 3-space given by the equation $z=2$.

• Describe the surface in 3-space given by $(x-4)(y-7)=0$.

Vectors in space

Except for concepts involving slope and and the $x$-axis angle $\theta$ ($\vec{v}=r\cos \theta \, \hat{\imath} +r\sin \theta \, \hat{\jmath}$), vectors in 2-space generalize nicely to vectors in 3-space. Vectors in 3-space have three components, and the basis vectors are now

As before, the basis vectors are the unit vectors in the directions of the coordinate axes.

Examples

• Determine a vector of magnitude $3$ whose direction is opposite that of the vector from $P\,(1,-2,4)$ to $Q\,(-2,-2,1)$.

• Let $\vec{u}=-3\hat{\imath}+2\hat{\jmath}+6\hat{k}$, $\vec{v}=4\hat{\jmath}-7\hat{k}$, and $\vec{w}=\langle 7,-4,-6 \rangle$. Find the magnitude of $2\vec{u}-\vec{v}+3\vec{w}$.

Parallel vectors

Vectors are parallel if they have the same or opposite directions. It is perhaps easiest to think about this idea in terms of scalar multiplication: Two vectors are parallel if and only if one is a scalar multiple of the other. That is

• Find the number $r$ so that $\vec{u}=6\hat{\imath}-3\hat{\jmath}+12\hat{k}$ and $\langle -4,r,-8 \rangle$ are parallel.

• Show that the points $A\,(1,0,1)$, $B\,(0,1,1)$, and $C\,(1,1,1)$ are not collinear.