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Lesson 26 — Vectors in the Plane

Vector as an object with magnitude, direction, and orientation. Addition, scalar multiplication, decomposition.

Used in: 1.º ano do EM (15–16 anos) · Equiv. Math I japonês §A — Vetores · Equiv. Klasse 11 alemã — Vektoren

v⃗=(v1,v2),∣v⃗∣=v12+v22,u⃗+v⃗=(u1+v1,u2+v2)\vec{v} = (v_1, v_2), \quad |\vec{v}| = \sqrt{v_1^2 + v_2^2}, \quad \vec{u} + \vec{v} = (u_1+v_1, u_2+v_2)
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Rigorous notation, full derivation, hypotheses

Vectors in ℝ²

Operations

  • Sum: u⃗+v⃗=(u1+v1,u2+v2)\vec u + \vec v = (u_1 + v_1, u_2 + v_2).
  • Scalar: αv⃗=(αv1,αv2)\alpha \vec v = (\alpha v_1, \alpha v_2).
  • Subtraction: u⃗−v⃗=(u1−v1,u2−v2)\vec u - \vec v = (u_1 - v_1, u_2 - v_2).

Magnitude (norm)

∣v⃗∣=v12+v22|\vec v| = \sqrt{v_1^2 + v_2^2}

Unit vector

v^=v⃗/∣v⃗∣\hat v = \vec v / |\vec v| has magnitude 1. Versor.

Canonical vectors

ı^=(1,0)\hat\imath = (1, 0), ȷ^=(0,1)\hat\jmath = (0, 1). Every vector: v⃗=v1ı^+v2ȷ^\vec v = v_1 \hat\imath + v_2 \hat\jmath.

Polar form

v⃗=(∣v⃗∣cos⁡θ,∣v⃗∣sin⁡θ)\vec v = (|\vec v|\cos\theta, |\vec v|\sin\theta) where θ\theta is the angle with the positive xx-axis.

Properties (8 vector-space axioms)

PropertyExpression
Commutativeu⃗+v⃗=v⃗+u⃗\vec u + \vec v = \vec v + \vec u
Associative(u⃗+v⃗)+w⃗=u⃗+(v⃗+w⃗)(\vec u + \vec v) + \vec w = \vec u + (\vec v + \vec w)
Identityv⃗+0⃗=v⃗\vec v + \vec 0 = \vec v
Inversev⃗+(−v⃗)=0⃗\vec v + (-\vec v) = \vec 0
Distributive (scalar/vector)α(u⃗+v⃗)=αu⃗+αv⃗\alpha(\vec u + \vec v) = \alpha\vec u + \alpha\vec v
Distributive (vector/scalar)(α+β)v⃗=αv⃗+βv⃗(\alpha + \beta)\vec v = \alpha\vec v + \beta\vec v
Compatibilityα(βv⃗)=(αβ)v⃗\alpha(\beta\vec v) = (\alpha\beta)\vec v
Scalar identity1⋅v⃗=v⃗1 \cdot \vec v = \vec v

These 8 properties characterize a vector space — seen formally in linear algebra (Term 12).

Parallelogram rule

u⃗+v⃗\vec u + \vec v is the diagonal of the parallelogram formed by u⃗\vec u and v⃗\vec v.

Exercise list

35 exercises · 8 with worked solution (25%)

Application 20Understanding 1Modeling 12Challenge 1Proof 1
  1. Ex. 26.1Application
    Compute (3,4)+(1,2)(3, 4) + (1, 2). (Ans: (4,6)(4, 6).)
    Solve online
  2. Ex. 26.2Application
    Compute 2⋅(3,−1)2 \cdot (3, -1). (Ans: (6,−2)(6, -2).)
    Solve online
  3. Ex. 26.3Application
    Compute (5,7)−(2,3)(5, 7) - (2, 3).
    Solve online
  4. Ex. 26.4ApplicationAnswer key
    Magnitude of (3,4)(3, 4). (Ans: 5.)
    Solve online
  5. Ex. 26.5Application
    Magnitude of (5,−12)(5, -12). (Ans: 13.)
    Solve online
  6. Ex. 26.6Application
    Unit vector in the direction of (6,8)(6, 8). (Ans: (3/5,4/5)(3/5, 4/5).)
    Solve online
  7. Ex. 26.7ApplicationAnswer key
    For u⃗=(1,2)\vec u = (1, 2), v⃗=(3,−1)\vec v = (3, -1): compute u⃗+v⃗\vec u + \vec v, 2u⃗−v⃗2\vec u - \vec v, ∣u⃗+v⃗∣|\vec u + \vec v|.
    Solve online
  8. Ex. 26.8Application
    Show that (3,4)(3, 4) and (−3,−4)(-3, -4) are opposites.
    Solve online
  9. Ex. 26.9Application
    Decompose v⃗=(5,5)\vec v = (5, 5) on the canonical basis (ı^,ȷ^)(\hat\imath, \hat\jmath).
    Solve online
  10. Ex. 26.10Application
    Vector with the same magnitude as (3,4)(3, 4) but opposite direction.
    Solve online
  11. Ex. 26.11ApplicationAnswer key
    Vector of magnitude 10 in the direction of (3,4)(3, 4). (Ans: (6,8)(6, 8).)
    Solve online
  12. Ex. 26.12Application
    Find v⃗\vec v such that v⃗+(2,−1)=(5,7)\vec v + (2, -1) = (5, 7).
    Solve online
  13. Ex. 26.13Application
    Show ∣αv⃗∣=∣α∣∣v⃗∣|\alpha \vec v| = |\alpha| |\vec v| for α∈R\alpha \in \mathbb{R}.
    Solve online
  14. Ex. 26.14Application
    Vector from A=(1,2)A = (1, 2) to B=(5,8)B = (5, 8) is AB⃗\vec{AB}. Compute. (Ans: (4,6)(4, 6).)
    Solve online
  15. Ex. 26.15Application
    Triangle A=(0,0)A = (0,0), B=(4,0)B = (4,0), C=(2,3)C = (2, 3). Compute AB⃗\vec{AB}, BC⃗\vec{BC}, CA⃗\vec{CA} and show that they sum to zero.
    Solve online
  16. Ex. 26.16Application
    Unit vector in the direction of the positive yy-axis: ȷ^=(0,1)\hat\jmath = (0, 1).
    Solve online
  17. Ex. 26.17Application
    For u⃗=(4,3)\vec u = (4, 3), find a perpendicular vector of the same magnitude. (Ans: (−3,4)(-3, 4) or (3,−4)(3, -4).)
    Solve online
  18. Ex. 26.18Application
    For which kk is ∣(k,3)∣=5|(k, 3)| = 5? (Ans: k=±4k = \pm 4.)
    Solve online
  19. Ex. 26.19Application
    Find α,β\alpha, \beta such that α(1,0)+β(0,1)=(3,7)\alpha(1, 0) + \beta(0, 1) = (3, 7).
    Solve online
  20. Ex. 26.20Application
    Linear combination w⃗=2u⃗−3v⃗\vec w = 2\vec u - 3\vec v with u⃗=(1,2)\vec u = (1,2), v⃗=(−1,1)\vec v = (-1, 1).
    Solve online
  21. Ex. 26.21Modeling
    In mechanics, force F⃗1=(3,4)\vec F_1 = (3, 4) N and F⃗2=(−1,2)\vec F_2 = (-1, 2) N act on a body. Resultant? (Ans: (2,6)(2, 6) N.)
    Solve online
  22. Ex. 26.22Modeling
    River with current c⃗=(3,0)\vec c = (3, 0) km/h, boat with motor m⃗=(0,4)\vec m = (0, 4) km/h. Resultant velocity. Does the trajectory leave the bank?
    Solve online
  23. Ex. 26.23Modeling
    Aviator at 500 km/h on heading 060°060° NE with 8080 km/h wind from the east. Resultant velocity (magnitude and angle).
    Solve online
  24. Ex. 26.24ModelingAnswer key
    Aircraft trajectory under 2 consecutive winds: v⃗1=(200,100)\vec v_1 = (200, 100) on the first leg, v⃗2=(300,−50)\vec v_2 = (300, -50) on the second. Time on each leg: 1h. Final position?
    Solve online
  25. Ex. 26.25Modeling
    In packet routing in a network, a hop vector is (lat, long, lat, long, ...) — model 3 consecutive hops.
    Solve online
  26. Ex. 26.26ModelingAnswer key
    In games, a player at (10,20)(10, 20) moves with velocity (5,−3)(5, -3) per second. Position after 4 s? (Ans: (30,8)(30, 8).)
    Solve online
  27. Ex. 26.27Modeling
    Embeddings in ML: word "king" ≈(0.3,0.5,0.2,...)\approx (0.3, 0.5, 0.2, ...), "queen" ≈(0.3,0.6,0.1,...)\approx (0.3, 0.6, 0.1, ...). Vector distance is semantic proximity.
    Solve online
  28. Ex. 26.28Modeling
    In GPS, your position is a 3D vector. Motion is a velocity vector. Instantaneous acceleration reported by the accelerometer: vector.
    Solve online
  29. Ex. 26.29ModelingAnswer key
    In statics, 3 cables pull point PP with forces F⃗1=(5,0)\vec F_1 = (5, 0), F⃗2=(−3,4)\vec F_2 = (-3, 4), F⃗3=(?,?)\vec F_3 = (?, ?). For equilibrium, F⃗3=?\vec F_3 = ?. (Ans: (−2,−4)(-2, -4).)
    Solve online
  30. Ex. 26.30ModelingAnswer key
    In 2D robotics, an arm with 2 segments. First segment in direction u⃗1=(cos⁡30°,sin⁡30°)⋅50\vec u_1 = (\cos 30°, \sin 30°) \cdot 50 cm. Second in direction u⃗2\vec u_2. Final position is u⃗1+u⃗2\vec u_1 + \vec u_2.
    Solve online
  31. Ex. 26.31Modeling
    Drone with 4 motors providing thrusts F⃗1,F⃗2,F⃗3,F⃗4\vec F_1, \vec F_2, \vec F_3, \vec F_4. To hover, the sum must offset gravity G⃗=(0,−mg)\vec G = (0, -mg).
    Solve online
  32. Ex. 26.32Modeling
    In quant finance, portfolio return is the linear combination r⃗=w1a⃗+w2b⃗\vec r = w_1 \vec a + w_2 \vec b of asset returns a⃗,b⃗\vec a, \vec b with weights w1,w2w_1, w_2.
    Solve online
  33. Ex. 26.33Understanding
    Show that if u⃗+v⃗=0⃗\vec u + \vec v = \vec 0, then v⃗=−u⃗\vec v = -\vec u.
    Solve online
  34. Ex. 26.34Challenge
    A vector v⃗\vec v has magnitude 10 and forms a 60°60° angle with the positive xx-axis. Components? (Ans: (5,53)(5, 5\sqrt 3).)
    Solve online
  35. Ex. 26.35ProofAnswer key
    Prove the triangle inequality ∣u⃗+v⃗∣≤∣u⃗∣+∣v⃗∣|\vec u + \vec v| \leq |\vec u| + |\vec v| via expansion of ∣u⃗+v⃗∣2|\vec u + \vec v|^2.

Sources

Updated on 2026-04-30 · Author(s): Clube da Matemática

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