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### Transcript of More about vectors. Coplanar vectors Q A B C a b c QA = λa QB = μb c = QA + QB c = λa + μb a, b,...

• Coplanar vectorsQABCQA = aQB = bc = QA + QBc = a + b

a, b, and c are coplanar vectorsa = (1, 1)b = (2, 1)c = (3, 4) = ? = ? 5-1

• Coplanar vector in 3Dzx21O12312334acbc = a + b

• Position vectorA vector which has its initial point at the origin of coordinates.xyyQxQrrxryQ (xQ, yQ)Ozx21O12312334rP23P (2, 3, 3)= (2, 3, 3)r = (xQ, yQ)

• Vector equations for curves & surfacesCircleyrCrC (a, b)OP (x, y)r - rCr rC = cProof: r rC = (x, y) (a, b) = (x-a, y-b)r - rC

• Vector equations for curves & surfacesPlanezxOACBAP = AB + AC

xrArBrCrPAB = rB - rAAC = rC - rAAP = r - rAr rA = (rB - rA)+ (rC rA)

-- Parametric vector equation for a plane

• Example-planeA (1,2,1), B(2,2,0), C(2,1,2)(x, y, z) (1, 2, 1) = {(2, 2, 0)-(1, 2, 1)} + {(2, 1, 2)-(1, 2, 1)}r rA = (rB - rA)+ (rC rA)= (1, 0, -1) + (1, -1, 1)

x = 1 + + y = 2 - z = 1 + x + 2 y + z = 6Cartesian parametric equations for the planeCartesian (general) equation for a plane(x, y, z) = (1++, 2-, 1-+)Vector parametric equation for a plane

• Plane contdGeneral form of a plane

A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3)

1c db da dx y z = + +cba

• Plane contdGeneral Cartesian form of a plane

A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3)

dx y z = + +cba

• Special planesGeneral form of a plane

Special cases:

d=0: ax+by+cz=0a plane passing through the origina=0: by+cz=da plane parallel to the x axisb=0: ax+cz=da plane parallel to the y axisc=0: ax+by=da plane parallel to the z axis

d

• Vector equations for curves & surfacesLinezxOCAAP = ACxrArCrPAC = rC - rAAP = r - rAr rA = (rC rA)-- Parametric vector equation for a line

• ExampleLineA(1,2,1), C(3, 0, -1)zxOCAxrArCrPr rA = (rC rA)(x, y, z)-(1, 2, 1)={(3, 0, -1)-(1, 2,1)}(x, y, z)=(1+2,2-2, 1-2)AP = AC

• Unit vectorA vector of unit magnitudee.g. What about the vectors, b=(1,0,0), c=(0,1,0), d=(0,0,1)?

• If a is a vector, thenthe unit vector in the direction of a =(ax, ay, az) is:azxyzxy

• Basis vectorsi = (1, 0, 0)j = (0, 1, 0)k = (0, 0, 1)Any vector, a = (a1, a2, a3) can be written as:a = a1 +a2 +a3xyz111

• 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