2.2 Vectors in Three Dimensions

← Chapter 2 §2.2 Vectors in Three Dimensions 空间向量 · Chapter 2 · Vectors in Space

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Access for free at openstax.org · CC BY-NC-SA 4.0

Objective	Takeaways; say them first, then click
2.2.1 Describe three-dimensional space mathematically.	three coordinates; right-handed axes; coordinate planes; eight octants
2.2.2 Locate points in space using coordinates.	$(x,y,z)$ : walk $x$ , $y$ , $z$ ; the far corner of the box
2.2.3 Write the distance formula in three dimensions.	$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$ : Pythagoras twice
2.2.4 Write the equations for simple planes and spheres.	a set of solutions; $x=a$ , $y=b$ , $z=c$ ; $(x-a)^2+(y-b)^2+(z-c)^2=r^2$
2.2.5 Perform vector operations in $\mathbb R^3$ .	$\langle x,y,z\rangle=x\mathbf i+y\mathbf j+z\mathbf k$ ; slot by slot

Mistake	Result	Fix
$x$ and $y$ swapped on the page	a left-handed frame; §2.4's cross products flip	right-hand rule first
$x=3$ read as a line	in space it is a plane	count the free coordinates
Radius of $\dots=173$ taken as 173	a sphere 13 times too big	$r=\sqrt{173}\approx13.15$
Centre of $(x+1)^2+(y-3)^2+(z-1)^2=21$ as $(1,-3,-1)$	every sign flipped	$(x+1)^2=(x-(-1))^2$ : $(-1,3,1)$
Diameter used as the radius	$\sqrt{84}\approx9.17$ in Example 2.15	half of it: $\sqrt{21}$
Initial minus terminal	$\langle 7,15,4\rangle$ in Example 2.18: backwards	terminal minus initial
$\\|\mathbf v+\mathbf w\\|=\\|\mathbf v\\|+\\|\mathbf w\\|$	lengths added as if in line	add components, then take the length
$-2^2=-4$ inside a length	$\sqrt{102}$ for $\\|\langle -2,9,5\rangle\\|$	$(-2)^2=4$ : $\sqrt{110}$
$\mathbf w$ used as the velocity in Example 2.20	length 28.87, not 60	$\mathbf v=\tfrac{60}{\\|\mathbf w\\|}\mathbf w$

#	task	answer
2.11	sketch $(-2,3,-1)$	octant $(-,+,-)$
2.12	distance $(1,-5,4)$ to $(4,-1,-1)$	$\sqrt{9+16+25}=5\sqrt2$
2.13	plane through $(1,-6,-4)$ parallel to $xy$	$z=-4$
2.14	centre $(-2,4,-5)$ , through $(4,4,-1)$	$(x+2)^2+(y-4)^2+(z+5)^2=52$
2.15	diameter $P(2,-1,-3)$ , $Q(-2,5,-1)$	$x^2+(y-2)^2+(z+2)^2=14$
2.16	$(y+2)(z-3)=0$	the planes $y=-2$ and $z=3$
2.17	$x^2+(z-2)^2=16$	cylinder, radius 4, axis $x=0,\ z=2$
2.18	$\overrightarrow{ST}$ , $S(3,8,2)$ , $T(2,-1,3)$	$\langle -1,-9,1\rangle=-\mathbf i-9\mathbf j+\mathbf k$
2.19	unit vector along $5\mathbf v+3\mathbf w$ , $\mathbf v=\langle -1,-1,1\rangle,\ \mathbf w=\langle 2,0,1\rangle$	$\tfrac{1}{3\sqrt{10}}\langle 1,-5,8\rangle$
2.20	the pass at 40 mph, 45°	$\langle 16\sqrt2,\ 12\sqrt2,\ 20\sqrt2\rangle$

#	task	answer	key step
61	box $O$ to $A(2,3,5)$ : other vertices, $OA$ ; p126	$(2,0,5),(2,0,0),(2,3,0),(0,3,0),(0,3,5),(0,0,5)$ ; $\sqrt{38}$	each coordinate 0 or A's
63	$(y-5)(z-6)=0$	planes $y=5$ and $z=6$	a factor is 0
67	through $(1,1,1)$ parallel to $xy$	$z=1$	fix $z$
71	centre $(-1,7,4)$ , radius 4	$(x+1)^2+(y-7)^2+(z-4)^2=16$	right side $r^2$
73	diameter $P(-1,5,7)$ , $Q(-5,2,9)$	$(x+3)^2+(y-3.5)^2+(z-8)^2=\tfrac{29}{4}$	midpoint; $r^2=\tfrac14(16+9+4)$
75	$x^2+y^2+z^2-4z+3=0$	centre $(0,0,2)$ , radius 1	complete the square
77	$\overrightarrow{PQ}$ , $P(3,0,2)$ , $Q(-1,-1,4)$ ; p127	$\langle -4,-1,2\rangle$	terminal minus initial
79	$P(-2,5,-8)$ , midpoint $M(1,-7,4)$	$\langle 6,-24,24\rangle$	$\overrightarrow{PQ}=2\overrightarrow{PM}$
83	$\mathbf a=\langle -1,-2,4\rangle,\ \mathbf b=\langle -5,6,-7\rangle$ : $\mathbf a+\mathbf b,\ 4\mathbf a,\ -5\mathbf a+3\mathbf b$	$\langle -6,4,-3\rangle$ , $\langle -4,-8,16\rangle$ , $\langle -10,28,-41\rangle$	component by component
87	$\\|\mathbf u-\mathbf v\\|,\ \\|-2\mathbf u\\|$ , $\mathbf u=2\mathbf i+3\mathbf j+4\mathbf k,\ \mathbf v=-\mathbf i+5\mathbf j-\mathbf k$	$\sqrt{38}$ , $2\sqrt{29}$	$\\|-2\mathbf u\\|=2\\|\mathbf u\\|$
93	unit vector along $\overrightarrow{PQ}$ , $P(-2,3,1)$ , $Q(0,-4,4)$	$\tfrac{1}{\sqrt{62}}\langle 2,-7,3\rangle$	the book's key prints $+7$ ; $-4-3=-7$
99	$\\|\mathbf u\\|=10$ along $\langle 7,-1,3\rangle$	$\tfrac{10}{\sqrt{59}}\langle 7,-1,3\rangle$	unit vector, then scale
103	length 5 along $\overrightarrow{AB}$ , $A(2,1,5)$ , $B(3,4,-7)$ ; p128	$\tfrac{5}{\sqrt{154}}\langle 1,3,-12\rangle$	the Example 2.20 move
111	50 N along $\overrightarrow{OP}$ , $P(3,4,0)$ ; angle with $+x$	$\langle 30,40,0\rangle$ ; $53^\circ$	$\cos\alpha=30/50$

Vectors in Three Dimensions

A point in space needs three coordinates, one per perpendicular axis

Right-hand rule: curl $x$ toward $y$ , the thumb is $z$

Each pair of axes spans a coordinate plane

The three coordinate planes cut space into eight octants

In space, lines that are not parallel can still miss each other

To plot $(x,y,z)$ , walk $x$ , then $y$ , then $z$

Set three coordinates: the point is the far corner of their box

Distance in space is Pythagoras twice

Subtract, square, add, take the root: $d=\sqrt{41}$

Any two points: across the floor first, then straight up

Two right triangles prove the distance formula

The same equation is a point, a line or a plane, depending on the space

A missing variable is free: a circle becomes a cylinder

A product equal to zero describes a union of surfaces

A plane parallel to a coordinate plane fixes one coordinate

Find the coordinate that stays fixed; that is the plane's equation

A sphere is every point at one distance from a centre

Radius from the distance formula; the equation keeps $r^2$

Fix a coordinate: a plane. Fix a distance: a sphere.

A diameter gives the centre (its midpoint) and the radius (half its length)

Complete the square to read the centre and the radius

A vector in space has three components: $\langle x,y,z\rangle=x\mathbf i+y\mathbf j+z\mathbf k$

Component form is terminal minus initial, in all three slots

A vector is the trip from $P$ to $Q$ , wherever it starts

Every vector operation works one coordinate at a time, as in the plane

Scale, then combine slot by slot; $\|5\mathbf w\|=5\|\mathbf w\|$

Scale each vector, then add slot by slot: $a\mathbf v+b\mathbf w$

A throw: direction from the field, length from the arm

Take the direction from geometry, then fix the length with a scalar

Space is the plane with one more coordinate

Common mistakes

Practice and answers