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More about vectors

Explore coplanar vectors, vector equations for curves and surfaces, parametric and Cartesian forms, special planes, and unit vectors.

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More about vectors

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  1. More about vectors

  2. c a b Coplanar vectors A QA = λa C QB = μb Q c = QA + QB B c = λa +μb a, b, and c are coplanar vectors a = (1, 1) b = (2, 1) c = (3, 4) λ = ? μ = ? 5 -1

  3. Coplanar vector in 3D z c = λa +μb 3 2 a 1 c 1 2 3 1 2 O b 3 4 x

  4. Position vector • A vector which has its initial point at the origin of coordinates. y Q (xQ, yQ) z yQ ry 3 P (2, 3, 3) 2 rP r 1 3 1 2 3 r = (xQ, yQ) 1 2 O 2 3 4 rx = (2, 3, 3) O x xQ x

  5. Vector equations for curves & surfaces • Circle r – rC = c y Proof: P (x, y) r – rC= (x, y) – (a, b) = (x-a, y-b) r - rC r r - rC C (a, b) rC O

  6. Vector equations for curves & surfaces • Plane AB = rB - rA z AC = rC - rA B A AP = r - rA rA rB P r C AP = λAB +μAC rC x O r – rA= λ (rB - rA)+μ(rC – rA) x -- Parametric vector equation for a plane

  7. Example-plane • A (1,2,1), B(2,2,0), C(2,1,2) r – rA= λ (rB - rA)+μ(rC – rA) (x, y, z) = (1+λ+μ, 2-μ, 1-λ+μ) (x, y, z) –(1, 2, 1) = λ{(2, 2, 0)-(1, 2, 1)}+ μ{(2, 1, 2)-(1, 2, 1)} = λ(1, 0, -1)+ μ(1, -1, 1) Vector parametric equation for a plane x = 1 + λ+ μ y = 2 - μ z = 1 – λ + μ x + 2 y + z = 6 Cartesian (general) equation for a plane Cartesian parametric equations for the plane

  8. Plane cont’d • General form of a plane • A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) a a’ x + b’ b y + c’ c z = d’ 1 d’ d’

  9. Plane cont’d • General Cartesian form of a plane • A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) a x + b y + c z = d

  10. a x + b y + c z = Special planes • General form of a plane • Special cases: • d=0: ax+by+cz=0 a plane passing through the origin • a=0: by+cz=d a plane parallel to the x axis • b=0: ax+cz=d a plane parallel to the y axis • c=0: ax+by=d a plane parallel to the z axis d

  11. Vector equations for curves & surfaces • Line z AC = rC - rA A AP = r - rA rA P r rC C AP = λAC O x r – rA= λ (rC – rA) x -- Parametric vector equation for a line

  12. Example—Line • A(1,2,1), C(3, 0, -1) AP = λAC r – rA= λ(rC – rA) z (x, y, z)-(1, 2, 1)=λ{(3, 0, -1)-(1, 2,1)} A (x, y, z)=(1+2λ,2-2λ, 1-2λ) rA P r C rC O x x

  13. Unit vector • A vector of unit magnitude • e.g. What about the vectors, b=(1,0,0), c=(0,1,0), d=(0,0,1)?

  14. a If a is a vector, then… • the unit vector in the direction of a=(ax, ay, az) is: z a θz θy θx y x

  15. k k j j i i Basis vectors • i = (1, 0, 0) • j = (0, 1, 0) • k = (0, 0, 1) • Any vector, a = (a1, a2, a3) can be written as: • a = a1 +a2 +a3 z 1 1 1 y x

  16. k j i a = (a1, a2, a3) = (a1, 0, 0)+(0, a2, 0)+(0, 0, a3) = a1(1, 0, 0)+a2 (0, 1, 0)+a3(0, 0, 1) = a1 +a2 +a3

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