Calculus Volume 3 · Chapter 2 · Section 2.3

The Dot Product

Two vectors in, one number out. It says how much they agree: positive, zero at a right angle, negative when they point apart.

Idea 1 · a numberMultiply matching components and add.⟨3, 5, 2⟩ · ⟨−1, 3, 0⟩ = 12, with no direction. Idea 2 · agreementu · v = ‖u‖ ‖v‖ cos θ.Its sign says acute, right or obtuse. Idea 3 · the shadowOne length times the other's shadow.The shadow is the projection; force along motion is work.

Colour key u v, F projection remainder q key idea · 3D figures turn when dragged · page links open the textbook

Level 1 · see itWhat the dot product measuresbefore any formula

How much of the pull moves the wagon?

A child pulls a wagon 50 ft with 8 lb, at 55° to the ground (p140).

229 ft·lb

of work — not 8 × 50 = 400. Only the part of the pull along the motion, 8 cos 55° ≈ 4.59 lb, moves the wagon.

$W=Fd$ assumes the force points along the motion (p143). The dot product counts only the part that agrees.

Red: the part along the ground, 4.59 lb. Dashed: the upright part, 6.55 lb, which does no work.

A number that measures agreement

A number. Multiply matching components and add: $\mathbf u\cdot\mathbf v=u_1v_1+u_2v_2+u_3v_3$ (p132).
Length × length × cos θ. $\mathbf u\cdot\mathbf v=\|\mathbf u\|\|\mathbf v\|\cos\theta$ (p134). Largest pointing the same way, zero at a right angle, negative pointing apart.
Length × shadow. $\|\mathbf v\|\cos\theta$ is the shadow of $\mathbf v$ on the line of $\mathbf u$ . The shadow, as a vector, is the projection.

Example 2.27a (p140): $\mathbf u=\langle -1,4,3\rangle$ , $\mathbf v=\langle 3,5,1\rangle$ . $\mathbf u\cdot\mathbf v=20$ , $\theta\approx48.47^\circ$ , shadow $20/\sqrt{26}\approx3.92$ , and $5.10\times3.92\approx20$ .

shadow and drop line angle θ

Red:

\operatorname{proj}_{\mathbf u}\mathbf v\approx\langle -0.77,3.08,2.31\rangle

. Amber dashed: the rest, at right angles to u. Drag to turn.

Positive, zero, negative

$\mathbf u=\langle 3,0\rangle$ stays; $\mathbf v$ , length 2.5, turns: $\mathbf u\cdot\mathbf v=7.5\cos\varphi$ .

turn of v, φ60°

Arc green: acute · grey: right · red: obtuse (p136).

The sign alone tells acute from obtuse — no inverse cosine needed.

angle θ	u · v	shadow of v on u
acute, $0\le\theta<\pi/2$	positive	along u
right, $\theta=\pi/2$	0	none
obtuse, $\pi/2<\theta\le\pi$	negative	against u

One picture

\|\mathbf u\|=4

\|\mathbf v\|=3

at 50°, 90°, 135°. Shadow forward: positive. No shadow: zero. Shadow backward: negative.

The dot product measures how much two vectors agree.

Level 2 · compute itAngles, projections, workand four labs

The definition and its rules

Dot product · p132

\mathbf u\cdot\mathbf v=u_1v_1+u_2v_2+u_3v_3 \tag{2.3}

A scalar. In the plane, drop the third term.

Theorem 2.3 (p132).

rule	name	means
$\mathbf u\cdot\mathbf v=\mathbf v\cdot\mathbf u$	commutative	agreement is mutual
$\mathbf u\cdot(\mathbf v+\mathbf w)=\mathbf u\cdot\mathbf v+\mathbf u\cdot\mathbf w$	distributive	shadows of a sum add
$c(\mathbf u\cdot\mathbf v)=(c\mathbf u)\cdot\mathbf v=\mathbf u\cdot(c\mathbf v)$	scalars factor out	double a vector, double the product
$\mathbf v\cdot\mathbf v=\\|\mathbf v\\|^2$	magnitude	a length squared is a dot product

$(\mathbf a\cdot\mathbf b)\,\mathbf c$ is a number times a vector; $\mathbf a\cdot(\mathbf b\cdot\mathbf c)$ means nothing.

Angles, right angles, direction cosines

Angle · p134

\cos\theta=\frac{\mathbf u\cdot\mathbf v}{\|\mathbf u\|\,\|\mathbf v\|} \tag{2.5}

$0\le\theta\le\pi$ : the smaller angle.

Orthogonal · p136

\mathbf u\perp\mathbf v\iff\mathbf u\cdot\mathbf v=0

For nonzero vectors; no lengths needed.

Direction cosines · p138

\langle\cos\alpha,\cos\beta,\cos\gamma\rangle=\frac{\mathbf v}{\|\mathbf v\|}

The angles with $\mathbf i,\mathbf j,\mathbf k$ .

Dividing by both lengths leaves pure direction: two vectors agree as much as their unit vectors do.

Projections and work

Vector projection · p140

\operatorname{proj}_{\mathbf u}\mathbf v=\frac{\mathbf u\cdot\mathbf v}{\|\mathbf u\|^2}\,\mathbf u \tag{2.6}

Scalar projection

\operatorname{comp}_{\mathbf u}\mathbf v=\frac{\mathbf u\cdot\mathbf v}{\|\mathbf u\|} \tag{2.7}

Work · p143

W=\mathbf F\cdot\overrightarrow{PQ} \tag{2.8}

lb × ft = ft·lb; N × m = J.

Decomposition (p141): $\mathbf v=\mathbf p+\mathbf q$ with $\mathbf p=\operatorname{proj}_{\mathbf u}\mathbf v$ and $\mathbf q=\mathbf v-\mathbf p\perp\mathbf u$ . One part along u, one at right angles; only the part along the motion does work.

Examples 2.21–2.30

Example	given	result
2.21 · p132	$\langle 3,5,2\rangle\cdot\langle -1,3,0\rangle$ ; $(10\mathbf i-4\mathbf j+7\mathbf k)\cdot(-2\mathbf i+\mathbf j+6\mathbf k)$	12; 18
2.22 · p133	$\mathbf a=\langle 1,2,-3\rangle$ , $\mathbf b=\langle 0,2,4\rangle$ , $\mathbf c=\langle 5,-1,3\rangle$	$(\mathbf a\cdot\mathbf b)\mathbf c=\langle -40,8,-24\rangle$ ; $\mathbf a\cdot(2\mathbf c)=-12$ ; $\\|\mathbf b\\|^2=20$
2.23 · p135	(a) $\mathbf i+\mathbf j+\mathbf k$ , $2\mathbf i-\mathbf j-3\mathbf k$ ; (b) $\langle 2,5,6\rangle$ , $\langle -2,-4,4\rangle$	(a) $\arccos(-2/\sqrt{42})\approx1.88$ rad; (b) $\pi/2$
2.24 · p136	$\langle 1,0,5\rangle$ , $\langle 10,3,-2\rangle$	orthogonal: $10+0-10=0$
2.25 · p137	direction angles of $\langle 2,3,3\rangle$	$\alpha\approx1.130$ , $\beta=\gamma\approx0.877$ rad
2.26 · p139	prices, costs, quantities as 4-vectors	sales $16,267.50; profit $14,383.70
2.27 · p140	(a) $\langle 3,5,1\rangle$ onto $\langle -1,4,3\rangle$ ; (b) $3\mathbf i-2\mathbf j$ onto $\mathbf i+6\mathbf j$	(a) $\langle -\tfrac{10}{13},\tfrac{40}{13},\tfrac{30}{13}\rangle$ ; (b) $-\tfrac{9}{37}\mathbf i-\tfrac{54}{37}\mathbf j$
2.28 · p141	$\langle 8,-3,-3\rangle$ along $\langle 2,3,2\rangle$	$\mathbf p=\tfrac{1}{17}\langle 2,3,2\rangle$ , $\mathbf q=\tfrac{1}{17}\langle 134,-54,-53\rangle$
2.29 · p142	20 knots; a 2-knot current 30° off the course	$20+2\cos30^\circ\approx21.73$ knots
2.30 · p144	$\mathbf F=5\mathbf i-3\mathbf j+\mathbf k$ N, $(1,1,1)\to(9,4,7)$ m	$\overrightarrow{PQ}=\langle 8,3,6\rangle$ , $W=37$ J

Also: $\langle 6,-4\rangle$ along $\langle 3,1\rangle$ splits into $\langle \tfrac{21}{5},\tfrac75\rangle+\langle \tfrac95,-\tfrac{27}{5}\rangle$ (Lab 2's start); the wagon does $8(50)\cos55^\circ\approx229$ ft·lb (p144).

A tiny dot product means nearly perpendicular: in Example 2.28 almost all of v is in q.

Lab 1 · the angle between two vectors

u₁1 u₂1 u₃1 v₁2 v₂−1 v₃−3

presets

shadow of v on u

Try: Ex 2.23b and Ex 2.24 — a right-angle mark appears. same way: cos θ = 1; opposite: −1.

Lab 2 · drag a projection

u₁3 u₂1 v₁6 v₂−4

presets

Drag the tips. Try: perpendicular — no shadow. obtuse — the shadow points backward. Lengthening u changes nothing: the projection depends only on u's line.

Lab 3 · direction angles in space

v₁2 v₂3 v₃3

presets

$\cos^2\alpha$ $\cos^2\beta$ $\cos^2\gamma$ always fill the bar.

Try: ⟨1, 1, 1⟩ — all three angles 54.74°. along j — the bar is all teal. The direction cosines forget the length.

Lab 4 · pull the wagon

force ‖F‖ (lb)8 angle θ55° distance (ft)50 trip so far (%)0%

Try: θ = 0° — 400 ft·lb. θ = 90° — none. Past 90° — negative: the force takes energy out. Checkpoint 2.29: 30 lb at 60° for 10 ft, 150 ft·lb.

Common mistakes

Mistake	Result	Fix
$\langle u_1v_1,u_2v_2,u_3v_3\rangle$	a vector where a number belongs	add the products
Not dividing by the lengths	$\arccos(-2)$ in Example 2.23(a)	$\cos\theta=\mathbf u\cdot\mathbf v/(\\|\mathbf u\\|\\|\mathbf v\\|)$
Mixing radians and degrees	"1.88°" for Example 2.23(a)	1.88 rad = 107.98°
$\\|\mathbf u\\|$ for $\\|\mathbf u\\|^2$ in (2.6)	the shadow $\\|\mathbf u\\|$ times too long	(2.6) has the square; (2.7) does not
$\operatorname{proj}_{\mathbf v}\mathbf u$ for $\operatorname{proj}_{\mathbf u}\mathbf v$	a vector along the wrong line	the subscript is the line projected onto
Endpoint for displacement	$\mathbf F\cdot\langle 9,4,7\rangle=40$ in Example 2.30	$\overrightarrow{PQ}=Q-P$ : 37 J
sin for work	$8(50)\sin55^\circ\approx328$ ft·lb	θ is measured from the motion: cos, 229
$\mathbf a\cdot(\mathbf b\cdot\mathbf c)$	a vector dotted with a number	only $(\mathbf a\cdot\mathbf b)\,\mathbf c$ makes sense
" $\mathbf u\cdot\mathbf v=0$ , so one is $\mathbf 0$ "	number rules applied to vectors	zero means perpendicular

Practice and answers

Exercises from p145; the book's key: p834–835, p840.

Checkpoints 2.21–2.29

#	task	answer
2.21	$\langle 2,9,-1\rangle\cdot\langle -3,1,-4\rangle$	$-6+9+4=7$
2.22	$\mathbf p=\langle 7,0,2\rangle,\mathbf q=\langle -2,2,-2\rangle,\mathbf r=\langle 0,2,-3\rangle$ : $(\mathbf r\cdot\mathbf p)\mathbf q$ ; $\\|\mathbf p\\|^2$	$-6\mathbf q=\langle 12,-12,12\rangle$ ; 53
2.23	angle between $\langle 1,2,0\rangle,\langle 2,4,1\rangle$	$\arccos(10/\sqrt{105})\approx0.22$ rad
2.24	x with $\langle 2,8,-1\rangle\perp\langle x,-1,2\rangle$	$2x-8-2=0\Rightarrow x=5$
2.25	direction angles of $\langle 3,-5,1\rangle$	$\alpha\approx1.04,\ \beta\approx2.58,\ \gamma\approx1.40$ rad
2.26	AAA in June: new prices and quantities	sales $15,685.50; profit $14,073.15
2.27	$5\mathbf i-\mathbf j$ along $4\mathbf i+2\mathbf j$	$\mathbf p=\tfrac{18}{5}\mathbf i+\tfrac95\mathbf j,\ \mathbf q=\tfrac75\mathbf i-\tfrac{14}{5}\mathbf j$
2.28	Example 2.29 with the current southeast	$20+2\cos60^\circ=21$ knots
2.29	30 lb at 60°, 10 ft	$30(10)\cos60^\circ=150$ ft·lb

Exercises

#	task	answer	key step
123	$\langle 3,0\rangle\cdot\langle 2,2\rangle$ (p145)	6	two terms in the plane
125	$\langle 2,2,-1\rangle\cdot\langle -1,2,2\rangle$	0	perpendicular
127	$(\mathbf a\cdot\mathbf b)\mathbf c$ , $(\mathbf a\cdot\mathbf c)\mathbf b$ for $\langle 2,0,-3\rangle,\langle -4,-7,1\rangle,\langle 1,1,-1\rangle$	$\langle -11,-11,11\rangle$ ; $\langle -20,-35,5\rangle$	number first, then scale
131	angle, $\langle 3,-1\rangle,\langle -4,0\rangle$	2.82 rad; not acute	$\cos\theta=-12/(4\sqrt{10})$
133	angle, $3\mathbf i,\ 4\mathbf i+4\mathbf j$	$\pi/4$ ; acute	$\cos\theta=12/(3\cdot4\sqrt2)$
137	angle, $\mathbf i+\mathbf j,\ \mathbf j-\mathbf k$	$\pi/3$	$\cos\theta=1/2$
143	orthogonal? $3\mathbf i-\mathbf j-2\mathbf k,\ -2\mathbf i-3\mathbf j+\mathbf k$ (p146)	no (−5)	Theorem 2.5
149	α with $2\mathbf i+3\mathbf j\perp9\mathbf i+\alpha\mathbf j$	$\alpha=-6$	$18+3\alpha=0$
153	angle A in triangle $A(1,1,8),B(4,-3,-4),C(-3,1,5)$	68.33°	$\overrightarrow{AB}\cdot\overrightarrow{AC}=24$ , lengths 13 and 5
161	direction cosines and angles of $\langle 2,2,1\rangle$ (p147)	$\tfrac23,\tfrac23,\tfrac13$ ; 48°, 48°, 71°	divide by $\\|\mathbf u\\|=3$
167	$\operatorname{proj}_{\mathbf u}\mathbf v$ , $\operatorname{comp}_{\mathbf u}\mathbf v$ : $\mathbf u=5\mathbf i+2\mathbf j,\ \mathbf v=2\mathbf i+3\mathbf j$	$\langle \tfrac{80}{29},\tfrac{32}{29}\rangle$ ; $\tfrac{16}{\sqrt{29}}$	(2.6) and (2.7)
171	decompose $3\mathbf i+2\mathbf j$ along $4\mathbf i-3\mathbf j$	$\langle \tfrac{24}{25},-\tfrac{18}{25}\rangle+\langle \tfrac{51}{25},\tfrac{68}{25}\rangle$	$\mathbf q=\mathbf v-\mathbf p$ , check $\mathbf p\cdot\mathbf q=0$
173	methane: distance PR; bond angle between $\overrightarrow{OS},\overrightarrow{OR}$	$2\sqrt2$ ; 109.47°	$\cos\theta=-1/3$
175	work, $\langle 5,6,-2\rangle$ N from $(3,-1,0)$ to $(2,3,1)$ m (p148)	17 N·m	$\overrightarrow{PQ}=\langle -1,4,1\rangle$
177	25 lb at 20°, 50 ft	1175 ft·lb	$25(50)\cos20^\circ$

Stretch: Exercise 179, a 500-lb wind at N30°E on a boat sailing north 100 ft: $500(100)\cos30^\circ\approx43{,}301.27$ ft·lb.

Level 3 · why it worksWhere the formulas come fromshort arguments

Why u · v = ‖u‖‖v‖ cos θ

The triangle with sides $\mathbf u$ , $\mathbf v$ , $\mathbf v-\mathbf u$ and the law of cosines (p134):

\|\mathbf v-\mathbf u\|^2=\|\mathbf u\|^2+\|\mathbf v\|^2-2\|\mathbf u\|\|\mathbf v\|\cos\theta

\|\mathbf v-\mathbf u\|^2=(\mathbf v-\mathbf u)\cdot(\mathbf v-\mathbf u)=\|\mathbf v\|^2-2\,\mathbf u\cdot\mathbf v+\|\mathbf u\|^2

Compare the right-hand sides. The dot product is the term that turns Pythagoras into the law of cosines; it vanishes exactly at a right angle.

Zero means perpendicular

\mathbf u\cdot\mathbf v=0\iff\cos\theta=0\iff\theta=\frac{\pi}{2}\qquad(\mathbf u,\mathbf v\neq\mathbf 0)

Exercise 158 · the diagonals of a rhombus

Direction cosines' squares add to 1

\cos^2\alpha+\cos^2\beta+\cos^2\gamma=\frac{v_1^2+v_2^2+v_3^2}{\|\mathbf v\|^2}=1

Example 2.25: $\tfrac{4}{22}+\tfrac{9}{22}+\tfrac{9}{22}=1$ . A vector cannot make small angles with all three axes at once.

Where the projection comes from

Write $\mathbf v=t\,\mathbf u+\mathbf q$ with $\mathbf q\perp\mathbf u$ and dot with $\mathbf u$ :

\mathbf u\cdot\mathbf v=t\,\|\mathbf u\|^2\quad\Longrightarrow\quad t=\frac{\mathbf u\cdot\mathbf v}{\|\mathbf u\|^2}

Why |cos θ| never exceeds 1

That parabola is never negative, so its discriminant is not positive:

4(\mathbf u\cdot\mathbf v)^2-4\|\mathbf u\|^2\|\mathbf v\|^2\le0\quad\Longrightarrow\quad|\mathbf u\cdot\mathbf v|\le\|\mathbf u\|\,\|\mathbf v\|

Equality only for parallel vectors. Example 2.27a: $20\le\sqrt{26}\sqrt{35}\approx30.17$ .

The proof never uses "3": for the price lists of Example 2.26, the dot product defines the angle.

Where it is used next

Where	The dot product does
§2.4 Cross product	a second product with length $\\|\mathbf u\\|\\|\mathbf v\\|\sin\theta$ : dot measures agreement, cross measures spread
§2.5 Planes	$\mathbf n\cdot(\mathbf r-\mathbf r_0)=0$ : every direction in the plane is perpendicular to the normal
§2.5 Distances	point to plane = the scalar projection onto the normal
§4.6 Directional derivatives	$\nabla f\cdot\mathbf u$ , the shadow of the gradient
§6.2 Line integrals	work along a curve adds up $\mathbf F\cdot d\mathbf r$

	Property	Rule
i	commutative	$\mathbf u\cdot\mathbf v=\mathbf v\cdot\mathbf u$
ii	distributive	$\mathbf u\cdot(\mathbf v+\mathbf w)=\mathbf u\cdot\mathbf v+\mathbf u\cdot\mathbf w$
iii	associative (with a scalar)	$c(\mathbf u\cdot\mathbf v)=(c\mathbf u)\cdot\mathbf v=\mathbf u\cdot(c\mathbf v)$
iv	property of magnitude	$\mathbf v\cdot\mathbf v=\\|\mathbf v\\|^2$

	invitations	party favors	decorations	food service
price $\mathbf p$	2.50	1.50	4.50	1.25
cost $\mathbf c$	0.25	0.25	0.50	0.20
quantity $\mathbf q$	1258	342	2426	1354

Objective	You can now
2.3.1 dot product	$u_1v_1+u_2v_2+u_3v_3$ , a number
2.3.2 perpendicular	test $\mathbf u\cdot\mathbf v=0$ ; angle from $\cos\theta=\frac{\mathbf u\cdot\mathbf v}{\\|\mathbf u\\|\\|\mathbf v\\|}$
2.3.3 direction cosines	$\cos\alpha=v_1/\\|\mathbf v\\|,\ \dots$ ; squares add to 1
2.3.4 projection	$\operatorname{proj}_{\mathbf u}\mathbf v=\frac{\mathbf u\cdot\mathbf v}{\\|\mathbf u\\|^2}\mathbf u$ : the shadow; $\mathbf v=\mathbf p+\mathbf q$
2.3.5 work	$W=\mathbf F\cdot\overrightarrow{PQ}=\\|\mathbf F\\|\\|\overrightarrow{PQ}\\|\cos\theta$

	$\mathbf u$	$\mathbf v$	product
$x$	3	−1	−3
$y$	5	3	15
$z$	2	0	0
add			12

How much of the pull moves the wagon?

A number that measures agreement

Positive, zero, negative

One picture

The definition and its rules

Angles, right angles, direction cosines

Projections and work

Examples 2.21–2.30

Lab 1 · the angle between two vectors

Lab 2 · drag a projection

Lab 3 · direction angles in space

Lab 4 · pull the wagon

Common mistakes

Practice and answers

Why u · v = ‖u‖‖v‖ cos θ

Zero means perpendicular

Direction cosines' squares add to 1

Where the projection comes from

Why |cos θ| never exceeds 1

Where it is used next

The Dot Product

When force and motion point different ways, only the part along the motion counts

Multiply matching components and add: the answer is a number

The dot product follows the rules of multiplication, and v⋅v\mathbf v\cdot\mathbf vv⋅v is a length squared

Each property is a rule of numbers, used slot by slot

(a⋅b) c(\mathbf a\cdot\mathbf b)\,\mathbf c(a⋅b)c is a vector; a⋅(2c)\mathbf a\cdot(2\mathbf c)a⋅(2c) and ∥b∥2\|\mathbf b\|^2∥b∥2 are numbers

The component sum equals length × length × cosine of the angle

Comparing the law of cosines with an expanded dot product gives Theorem 2.4

∣u⋅v∣≤∥u∥ ∥v∥|\mathbf u\cdot\mathbf v|\le\|\mathbf u\|\,\|\mathbf v\|∣u⋅v∣≤∥u∥∥v∥, so the cosine formula never gives more than 1

i+j+k\mathbf i+\mathbf j+\mathbf ki+j+k and 2i−j−3k2\mathbf i-\mathbf j-3\mathbf k2i−j−3k meet at about 1.88 rad, an obtuse angle

The sign of u⋅v\mathbf u\cdot\mathbf vu⋅v tells acute, right or obtuse at a glance

Nonzero vectors are perpendicular exactly when their dot product is zero

With both lengths nonzero, u⋅v=0\mathbf u\cdot\mathbf v=0u⋅v=0 forces cos⁡θ=0\cos\theta=0cosθ=0

A vector's angles with the three axes have cosines v1/∥v∥, v2/∥v∥, v3/∥v∥v_1/\|\mathbf v\|,\ v_2/\|\mathbf v\|,\ v_3/\|\mathbf v\|v1​/∥v∥, v2​/∥v∥, v3​/∥v∥

A negative component gives an obtuse direction angle

Any list of numbers is a vector: price · quantity is the sales total

The projection of v\mathbf vv onto u\mathbf uu is its shadow on the line of u\mathbf uu

Projecting ⟨3,5,1⟩\langle 3,5,1\rangle⟨3,5,1⟩ onto ⟨−1,4,3⟩\langle -1,4,3\rangle⟨−1,4,3⟩ scales u\mathbf uu by 2026\tfrac{20}{26}2620​

A negative dot product makes the projection point against u\mathbf uu

Every vector splits into a part along u\mathbf uu and a part perpendicular to u\mathbf uu

The projection is the point of the line nearest to the head of v\mathbf vv

A tiny dot product means v\mathbf vv is nearly perpendicular to u\mathbf uu: almost all of it is q\mathbf qq

A current helps the ship only by its shadow on the course: 1.73 of its 2 knots

Work is force along the motion times distance: W=F⋅PQ→W=\mathbf F\cdot\overrightarrow{PQ}W=F⋅PQ​

Work falls to zero at 90° and turns negative beyond

With vectors given, work is one dot product: 37 J for the suitcase

The dot product measures how much two vectors agree

The dot product follows the rules of multiplication, and $\mathbf v\cdot\mathbf v$ is a length squared

$(\mathbf a\cdot\mathbf b)\,\mathbf c$ is a vector; $\mathbf a\cdot(2\mathbf c)$ and $\|\mathbf b\|^2$ are numbers

$|\mathbf u\cdot\mathbf v|\le\|\mathbf u\|\,\|\mathbf v\|$ , so the cosine formula never gives more than 1

$\mathbf i+\mathbf j+\mathbf k$ and $2\mathbf i-\mathbf j-3\mathbf k$ meet at about 1.88 rad, an obtuse angle

The sign of $\mathbf u\cdot\mathbf v$ tells acute, right or obtuse at a glance

With both lengths nonzero, $\mathbf u\cdot\mathbf v=0$ forces $\cos\theta=0$

A vector's angles with the three axes have cosines $v_1/\|\mathbf v\|,\ v_2/\|\mathbf v\|,\ v_3/\|\mathbf v\|$

The projection of $\mathbf v$ onto $\mathbf u$ is its shadow on the line of $\mathbf u$

Projecting $\langle 3,5,1\rangle$ onto $\langle -1,4,3\rangle$ scales $\mathbf u$ by $\tfrac{20}{26}$

A negative dot product makes the projection point against $\mathbf u$

Every vector splits into a part along $\mathbf u$ and a part perpendicular to $\mathbf u$

The projection is the point of the line nearest to the head of $\mathbf v$

A tiny dot product means $\mathbf v$ is nearly perpendicular to $\mathbf u$ : almost all of it is $\mathbf q$

Work is force along the motion times distance: $W=\mathbf F\cdot\overrightarrow{PQ}$