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Basics of Analytical Geometry

Basics of Analytical Geometry. By Kishore Kulkarni. Outline. 2D Geometry Straight Lines, Pair of Straight Lines Conic Sections Circles, Ellipse, Parabola, Hyperbola 3D Geometry Straight Lines, Planes, Sphere, Cylinders Vectors 2D & 3D Position Vectors

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Basics of Analytical Geometry

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  1. Basics of Analytical Geometry By Kishore Kulkarni

  2. Outline • 2D Geometry • Straight Lines, Pair of Straight Lines • Conic Sections • Circles, Ellipse, Parabola, Hyperbola • 3D Geometry • Straight Lines, Planes, Sphere, Cylinders • Vectors • 2D & 3D Position Vectors • Dot Product, Cross Product & Box Product • Analogy between Scalar and vector representations

  3. 2D Geometry • Straight Line • ax + by + c = 0 • y = mx + c, m is slope and c is the y-intercept. • Pair of Straight Lines • ax2 + by2 + 2hxy + 2gx + 2fy + c = 0 where abc + 2fgh – af2 – bg2 – ch2 = 0

  4. Conic Sections • Circle, Parabola, Ellipse, Hyperbola • Circle – Section Parallel to the base of the cone • Parabola - Section inclined to the base of the cone and intersecting the base of the cone • Ellipse - Section inclined to the base of the cone and not intersecting the base of the cone • Hyperbola – Section Perpendicular to the base of the cone

  5. Conic Sections • Circle: x2 + y2 = r2 , r => radius of circle • Parabola: y2 = 4ax or x2 = 4ay • Ellipse: x2/a2 + y2/b2 =1, a is major axis & b is minor axis • Hyperbola: x2/a2 - y2/b2 =1. In all the above equation, center is the origin. Replacing x by x-h and y by y-k, we get equations with center (h,k)

  6. Conic Sections • In general, any conic section is given by ax2 + by2 + 2hxy + 2gx + 2fy + c = 0 where abc + 2fgh – af2 – bg2 – ch2 != 0 • Special cases • h2 = ab, it is a parabola • h2 < ab, it is an ellipse • h2 > ab, it is a hyperbola • h2 < ab and a=b, it is a circle

  7. 3D Geometry • Plane - ax + by + cz + d = 0 • Sphere - x2 + y2 + z2 = r2 (x-h)2 + (y-k)2 + (z-l)2 = r2 , if center is (h, k, l) • Cylinder - x2 + y2 = r2, r is radius of the base. (x-h)2 + (y-k)2 = r2 , if center is (h, k, l)

  8. 3D Geometry • Question What region does this inequality represent in a 3D space ? 9 < x2 + y2 + z2 < 25

  9. 3D Geometry • Straight Lines • Parametric equations of line passing through (x0, y0, z0) x = x0 + at, y = y0 + bt, z = z0 + ct • Symmetric form of line passing through (x0, y0, z0) (x - x0)/a = (y - y0)/b = (z - z0)/c where a, b, c are the direction numbers of the line.

  10. Vectors • Any point in P in a 2D plane or 3D space can be represented by a position vector OP, where O is the origin. • Hence P(a,b) in 2D corresponds to position vector < a, b> and Q(a, b, c) in 3D space corresponds to position vector < a, b, c> • Let P <x1, y1, z1> and Q < x2, y2, z2 > then vector PQ = OQ – OP =< x2 – x1, y2 – y1, z2 – z1> • Length of a vector v = < v1, v2, v3> is given by |v| = sqrt(v12 + v22 + v32)

  11. Dot (Scalar) Product of vectors • Dot product of two vectors a = a1i + a2j + a3k and b = b1i + b2j + b3k is defined as a.b = a1b1 + a2b2 + a3b3. • Dot Product of two vectors is a scalar. • If θ is the angle between a and b, we can write a.b = |a||b|cosθ • Hence a.b = 0 implies two vectors are orthogonal. • Further a.b > 0 we can say that they are in the same general direction and a.b < 0 they are in the opposite general direction. • Projection of vector b on a = a.b / |a| • Vector Projection of vector b on a = (a.b / |a|) ( a / |a|)

  12. Direction Angles and Direction Cosines • Direction Angles α, β, γ of a vector a = a1i + a2j + a3k are the angles made by a with the positive directions of x, y, z axes respectively. • Direction cosines are the cosines of these angles. We have cos α = a1/ |a|, cos β = a2/ |a|, cos γ = a3/ |a|. • Hence cos2α + cos2β + cos2γ = 1. • Vector a = |a| <cos α, cos β, cos γ>

  13. Cross (Vector) Product of vectors • Cross product of two vectors a = a1i + a2j + a3k and b = b1i + b2j + b3k is defined as a x b = (a2b3 – a3b2)i +(a3b1 – a1b3)j +(a1b2 – a2b1)k. • a x b is a vector. • a x b is perpendicular to both a and b. • | a x b |=|a| |b| sinθ represents area of parallelogram.

  14. Cross (Vector) Product • Question What can you say about the cross product of two vectors in 2D ?

  15. Box Product of vectors • Box Product of vectors a, b and c is defined as V = a.(b x c) • Box Product is also called Scalar Tripple Product • Box product gives the volume of a parallelepiped.

  16. Vector Equations • Equation of a line L with a point P(x0, y0, z0) is given by r = r0 + tv where r0 = < x0, y0, z0>, r= < x, y, z>, v = <a, b, c> is a vector parallel to L, t is a scalar. • Equation of a plane is given by n.(r - r0) = 0 where n is a normal vector, which is analogous to the scalar equation a (x- x0) + b (y- y0) + c (z- z0) = 0

  17. Vector Equations • Let a and b be position vectors of points A(x1, y1,z1) and B(x2, y2,z2). Then position vector of the point P dividing the vector AB in the ratio m:n is given by p = (mb + na) / (m+n) which corresponds to P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n), (mz2 + nz1)/(m+n))

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