2.3 The Dot Product

← Chapter 2 §2.3 The Dot Product 点积 · Chapter 2 · Vectors in Space

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

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

	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

	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	$\mathbf v\cdot\mathbf v=\\|\mathbf v\\|^2$

Objective	Takeaways; say them first, then click
2.3.1 Calculate the dot product of two given vectors.	$u_1v_1+u_2v_2+u_3v_3$ , a number; rules of multiplication, $\mathbf v\cdot\mathbf v=\\|\mathbf v\\|^2$
2.3.2 Determine whether two given vectors are perpendicular.	$\\|\mathbf u\\|\\|\mathbf v\\|\cos\theta$ ; sign: acute, right, obtuse; nonzero: $\mathbf u\perp\mathbf v\iff\mathbf u\cdot\mathbf v=0$
2.3.3 Find the direction cosines of a given vector.	$\cos\alpha=v_1/\\|\mathbf v\\|,\ \dots$ ; squares add to 1
2.3.4 Explain and compute the vector projection.	the shadow $\frac{\mathbf u\cdot\mathbf v}{\\|\mathbf u\\|^2}\mathbf u$ ; $\mathbf v=\mathbf p+\mathbf q$ , $\mathbf p\perp\mathbf q$
2.3.5 Calculate the work done by a given force.	$W=\mathbf F\cdot\overrightarrow{PQ}$ ; zero at 90°, negative beyond

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

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

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

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

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

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

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

Set any $\mathbf u$ and $\mathbf v$ : the sign of $\mathbf u\cdot\mathbf v$ matches the angle you see

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

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

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

Nonzero vectors are perpendicular exactly when their dot product is zero

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

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

A negative component gives an obtuse direction angle

Change $\mathbf v$ : the squares of its direction cosines always add to 1

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 $\mathbf u$

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

Drag $\mathbf u$ and $\mathbf v$ : the projection depends only on the line of $\mathbf u$

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

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: $W=\mathbf F\cdot\overrightarrow{PQ}$

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

Work falls to zero at 90° and turns negative beyond

Pull the wagon: the work follows the cosine of the pull angle

The dot product measures how much two vectors agree

Common mistakes

Practice and answers

#	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

#	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	$25(50)\cos20^\circ$

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

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

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

(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

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

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

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

Set any u\mathbf uu and v\mathbf vv: the sign of u⋅v\mathbf u\cdot\mathbf vu⋅v matches the angle you see

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

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