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Lesson 30 — Term 3 Consolidation: Analytic Geometry + Vectors

Integration workshop for Lessons 21-29: analytic geometry, conics, vectors, linear systems.

Used in: 1.º ano do EM (16 anos) · Equiv. Math II japonês — geometria analítica plana · Equiv. Klasse 10/11 alemã — Analytische Geometrie

d^2 = \Delta x^2 + \Delta y^2, \quad y = mx + n, \quad \vec u \cdot \vec v = u_1 v_1 + u_2 v_2, \quad A\mathbf x = \mathbf b

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Rigorous notation, full derivation, hypotheses

Term 3 synthesis

Map of key formulas

Topic	Formula	Lessons
Distance		21
Midpoint		21
Line equation		22
Slope		22
Parallels		23
Perpendiculars		23
Point-line distance		23
Circle		24
Ellipse		25
Parabola		25
Hyperbola		25
Vector sum	component-wise	26
Vector magnitude		26
Dot product		27
Work		28
Newton's 2nd		28
Cramer 2x2		29

Problem styles (35 exercises)

Combined application (~15): geometry + algebra.
Modeling (~10): vector physics, telecom, trajectories, finance.
Challenge (~7): Brazilian Olympiad / ITA / IME level.
Proof (~3): consolidate rigorous proof.

Exercise list

35 exercises · 8 with worked solution (25%)

Application 12Modeling 13Challenge 5Proof 5

Ex. 30.1Application
Distance between $(2, 5)$ and $(8, 13)$ . (Ans: 10.)
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Ex. 30.2Application
Equation of the line through $(0, 4)$ parallel to $y = 2x - 3$ .
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Ex. 30.3Application
Perpendicular bisector of $\overline{(0,0)(6,0)}$ — equation. (Ans: $x = 3$ .)
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Ex. 30.4Application
Center of the circle $x^2 + y^2 - 6x + 4y - 12 = 0$ . (Ans: $(3, -2)$ .)
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Ex. 30.5Application
Position of $(5, 5)$ relative to $x^2 + y^2 = 25$ . (Ans: exterior, dist $5\sqrt 2 > 5$ .)
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Ex. 30.6ApplicationAnswer key
Equation of the parabola with focus $(3, 0)$ and directrix $x = -3$ . (Ans: $y^2 = 12x$ .)
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Ex. 30.7Application
Eccentricity of $x^2/16 + y^2/9 = 1$ .
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Ex. 30.8ApplicationAnswer key
Tangent to $x^2 + y^2 = 25$ at $(3, 4)$ .
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Ex. 30.9Modeling
Police siren: 500m radius coverage. Boundary equation in a system with the siren at $(0,0)$ .
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Ex. 30.10Modeling
4G antenna covers 2km. For 4 antennas at $(0,0), (3,0), (0,3), (3,3)$ , which region has multiple coverage?
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Ex. 30.11Application
$(2, -1) + 3 \cdot (1, 2)$ . (Ans: $(5, 5)$ .)
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Ex. 30.12Application
Angle between $(3, 4)$ and $(1, 0)$ .
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Ex. 30.13ApplicationAnswer key
Projection of $(5, 12)$ onto $(0, 1)$ . (Ans: $(0, 12)$ .)
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Ex. 30.14Modeling
Work of $\vec F = (8, 6)$ N to displace an object $\vec d = (3, 0)$ m. (Ans: 24 J.)
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Ex. 30.15Modeling
Mass of 5 kg on a 30° ramp. Sliding acceleration (no friction). (Ans: $g/2 \approx 4.9$ m/s².)
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Ex. 30.16Modeling
Plane at 700 km/h heading NE with 100 km/h east wind. Resultant velocity.
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Ex. 30.17ModelingAnswer key
Two cables support a 200 kg mass, at $30°$ and $45°$ angles from vertical. Tension in each cable.
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Ex. 30.18Modeling
In DSP, cosine similarity between $(0.5, 0.8, 0.3)$ and $(0.3, 0.6, 0.4)$ .
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Ex. 30.19ModelingAnswer key
In a Transformer, attention score for $Q = (1, 1)$ , $K = (3, 4)$ is $Q \cdot K / \sqrt 2$ .
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Ex. 30.20Modeling
Ballistic launch: $v_0 = 30$ m/s at $45°$ . Horizontal range? (Ans: $\approx 91.7$ m.)
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Ex. 30.21Application
Solve $\begin{cases} x + 2y = 7 \\ 3x - y = 1 \end{cases}$ . (Ans: $(\frac{9}{7}, \frac{20}{7})$ .)
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Ex. 30.22Modeling
Mixture: 30% and 60% to obtain 100 L at 45%. Find. (Ans: 50L of each.)
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Ex. 30.23Modeling
Small shop sells 2 products. 5A + 3B = $110, 2A + 4B = $80. Prices. (Ans: A=10, B=15.)
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Ex. 30.24Modeling
GPS triangulation: 3 satellites at $(0,0), (10,0), (0,10)$ measure distances $\sqrt{50}, \sqrt{50}, \sqrt{50}$ . Your position?
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Ex. 30.25Modeling
Markowitz with 2 assets: minimize $w_1^2 + 2w_2^2$ subject to $w_1 + w_2 = 1$ . Optimal weights? (Ans: $w_1 = 2/3, w_2 = 1/3$ .)
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Ex. 30.26Challenge
Largest circle inscribed in the triangle with vertices $(0,0), (10,0), (0,6)$ .
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Ex. 30.27Challenge
Find the ellipse through $(0, 0), (4, 0), (2, 3)$ with horizontal major axis.
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Ex. 30.28Challenge
Line $r$ tangent to the circle $x^2 + y^2 = 4$ parallel to $y = x + 1$ . Equation?
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Ex. 30.29Challenge
A vector $\vec v$ has magnitude $\sqrt{20}$ , is perpendicular to $(3, 1)$ , and has $v_2 > 0$ . Components?
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Ex. 30.30ChallengeAnswer key
System $\begin{cases} ax + y = 1 \\ x + ay = 1 \end{cases}$ — for which $a$ does it have (a) unique solution, (b) infinitely many solutions, (c) none?
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Ex. 30.31Proof
Prove that in a right triangle, the hypotenuse is the diameter of the circumscribed circle (Thales).
Ex. 30.32Proof
Prove Cauchy-Schwarz for $\mathbb{R}^2$ using the dot product.
Ex. 30.33ProofAnswer key
Prove the point-line distance formula using vector projection.
Ex. 30.34ProofAnswer key
Show that the perpendicular bisector of a segment is perpendicular to the segment and passes through the midpoint.
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Ex. 30.35Proof
Prove that the sum of the side vectors of any closed polygon is zero.

Sources

A consolidation lesson combines the sources from Lessons 21-29:

College Algebra — OpenStax · 2022, 2nd ed · EN · CC-BY · §2, §10.
Algebra and Trigonometry — OpenStax · 2022, 2nd ed · EN · CC-BY · §10.
Linear Algebra Done Right — Sheldon Axler · 2024, 4th ed · EN · CC-BY-NC · ch. 1, 6.
A First Course in Linear Algebra — Robert A. Beezer · 2022 · EN · GFDL · ch. V, O, SLE.
University Physics (Volume 1) — OpenStax · 2016 · EN · CC-BY · ch. 2-7: vector kinematics and dynamics.
1990 Nobel Prize in Economic Sciences — Markowitz, Miller, Sharpe · modern portfolio theory.
1997 Nobel Prize in Economic Sciences — Merton, Scholes · derivatives pricing.

Full catalog at /livros.