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Lesson 27 — Dot Product

Inner product (dot product). Angle between vectors, projection, orthogonality. Mechanical work.

Used in: 1.º ano EM (15 anos) · Equiv. Math II japonês · Equiv. Klasse 11 alemã · Precalculus §11.8 (US)

\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 = |\vec{u}||\vec{v}|\cos\theta

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Rigorous notation, full derivation, hypotheses

Definition and properties

Properties

Commutative: $\vec u \cdot \vec v = \vec v \cdot \vec u$ .
Distributive: $\vec u \cdot (\vec v + \vec w) = \vec u \cdot \vec v + \vec u \cdot \vec w$ .
Linear in scalar: $(\alpha \vec u) \cdot \vec v = \alpha (\vec u \cdot \vec v)$ .
Positive: $\vec u \cdot \vec u = |\vec u|^2 \geq 0$ , with equality $\iff \vec u = \vec 0$ .

Sign and geometry

$\vec u \cdot \vec v$	$\cos\theta$	Angle
$> 0$	$> 0$	acute $\theta < 90°$
$= 0$	$0$	right, $\theta = 90°$ (orthogonal)
$< 0$	$< 0$	obtuse $\theta > 90°$

Orthogonality

$\vec u \perp \vec v \iff \vec u \cdot \vec v = 0$ .

Angle

$\cos\theta = \frac{\vec u \cdot \vec v}{|\vec u| |\vec v|}$

Projection

Projection of $\vec u$ onto the direction of $\vec v$ : $\text{proj}_{\vec v} \vec u = \frac{\vec u \cdot \vec v}{|\vec v|^2} \vec v$

Magnitude (scalar) of the projection: $\frac{\vec u \cdot \vec v}{|\vec v|}$ .

Central application — mechanical work

$W = \vec F \cdot \vec d$ — work done by a force is its dot product with the displacement.

Exercise list

35 exercises · 8 with worked solution (25%)

Application 20Modeling 12Challenge 2Proof 1

Ex. 27.1ApplicationAnswer key
$(3, 4) \cdot (1, 2)$ . (Ans: 11.)
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Ex. 27.2Application
$(2, -1) \cdot (3, 5)$ . (Ans: 1.)
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Ex. 27.3Application
$(0, 0) \cdot \vec v$ for any $\vec v$ . (Ans: 0.)
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Ex. 27.4Application
Verify whether $(3, 4)$ and $(-4, 3)$ are perpendicular. (Ans: yes, dot = 0.)
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Ex. 27.5Application
For which $k$ is $(2, k) \cdot (3, 1) = 0$ ? (Ans: $k = -6$ .)
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Ex. 27.6ApplicationAnswer key
Angle between $(1, 0)$ and $(1, 1)$ . (Ans: 45°.)
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Ex. 27.7Application
Angle between $(3, 4)$ and $(4, 3)$ .
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Ex. 27.8Application
Show $|\vec v|^2 = \vec v \cdot \vec v$ for $\vec v = (2, 3)$ .
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Ex. 27.9Application
Projection of $(4, 3)$ onto $(1, 0)$ . (Ans: $(4, 0)$ .)
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Ex. 27.10Application
Projection of $(4, 3)$ onto $(0, 1)$ .
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Ex. 27.11Application
Projection of $(3, 5)$ onto $(1, 1)$ .
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Ex. 27.12Application
Decomposition of $(3, 5)$ into parallel + perpendicular to $(1, 0)$ .
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Ex. 27.13ApplicationAnswer key
For $\vec u = (1, 2), \vec v = (3, -1)$ : angle between them?
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Ex. 27.14ApplicationAnswer key
Unit vector orthogonal to $(2, 1)$ . (Ans: $(\pm 1, \mp 2)/\sqrt 5$ .)
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Ex. 27.15Application
Find a vector of magnitude 5 perpendicular to $(3, 4)$ .
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Ex. 27.16Application
Cosine of the angle between $(1, 0)$ and $(0, 1)$ . (Ans: 0.)
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Ex. 27.17ApplicationAnswer key
$\vec u \cdot \vec u$ is always non-negative. Prove.
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Ex. 27.18Application
For $\vec u = (3, 0), \vec v = (0, 4)$ : $\vec u \cdot \vec v = ?$ .
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Ex. 27.19Application
For $\vec u = (2, 3), \vec v = (-3, 2)$ : orthogonal? Angle? (Ans: yes, 90°.)
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Ex. 27.20Application
For which $\theta$ between non-zero vectors is $\vec u \cdot \vec v < 0$ ?
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Ex. 27.21Modeling
Work done by force $\vec F = (10, 5)$ N over displacement $\vec d = (3, 4)$ m: $W = \vec F \cdot \vec d$ . (Ans: 50 J.)
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Ex. 27.22ModelingAnswer key
Force $\vec F = (5, 0)$ N pulls a box through $\vec d = (3, 4)$ m. Useful work = projection of $\vec F$ onto $\vec d$ times $|\vec d|$ .
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Ex. 27.23Modeling
On a ramp, gravitational force $\vec g = (0, -mg)$ projected onto the ramp direction $(\cos\theta, -\sin\theta)$ . Component parallel to the plane = $mg \sin\theta$ .
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Ex. 27.24Modeling
In ML, cosine similarity between two embeddings: $\cos\theta = \vec u \cdot \vec v / (|\vec u||\vec v|)$ . For $(0.3, 0.5)$ and $(0.6, 0.4)$ , compute.
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Ex. 27.25Modeling
In recommendation, two users have rating vectors $(5,4,3,5,2)$ and $(4,5,3,4,3)$ . Cosine?
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Ex. 27.26Modeling
In a digital filter, correlation between signal $(1, 2, 1, 0)$ and template $(1, 1, 0, 0)$ via dot product. (Ans: 3.)
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Ex. 27.27Modeling
Non-trivial work: a force perpendicular to motion does zero work ( $\theta = 90°$ , $\cos = 0$ ).
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Ex. 27.28Modeling
Lambert's law (illumination): intensity $I = I_0 \vec n \cdot \hat\ell$ — dot product of normal with light direction.
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Ex. 27.29Modeling
In GPS, projection of radial error onto tangential direction via dot product.
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Ex. 27.30Modeling
In a Transformer (attention mechanism), score = $Q \cdot K / \sqrt d$ . For $Q = (1, 2), K = (3, 4), d = 2$ , compute.
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Ex. 27.31ModelingAnswer key
In quant finance, portfolio return is $r_p = \vec w \cdot \vec r$ with weights $\vec w$ and returns $\vec r$ . For $\vec w = (0.5, 0.3, 0.2)$ and $\vec r = (0.10, 0.08, -0.02)$ , compute.
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Ex. 27.32Modeling
Electric flux $\Phi = \vec E \cdot \vec A$ . For $\vec E = (5, 3)$ N/C and $\vec A = (0.2, 0.1)$ m², compute.
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Ex. 27.33ChallengeAnswer key
Prove the Cauchy-Schwarz inequality $|\vec u \cdot \vec v| \leq |\vec u||\vec v|$ . (Use $|\vec u + t\vec v|^2 \geq 0$ for all $t$ .)
Ex. 27.34Proof
Prove the vector law of cosines: $|\vec u - \vec v|^2 = |\vec u|^2 + |\vec v|^2 - 2 \vec u \cdot \vec v$ .
Ex. 27.35Challenge
Show that $\vec u \cdot \vec v = u_1 v_1 + u_2 v_2 = |\vec u||\vec v|\cos\theta$ using the law of cosines.
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Sources

Linear Algebra Done Right — Sheldon Axler · 2024, 4th ed · EN · CC-BY-NC · ch. 6: inner products. Primary source.
A First Course in Linear Algebra — Robert A. Beezer · 2022 · EN · GFDL · ch. O: orthogonality.
University Physics (Volume 1) — OpenStax · 2016 · EN · CC-BY · ch. 7: mechanical work. Source for Block B.
Mathematics for Machine Learning — Deisenroth, Faisal, Ong · 2020 · EN · free · ch. 3: dot product and similarity.