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Analytical Geometry Essentials: 2D, 3D, Vectors, Projections

Explore the fundamentals of Analytical Geometry including 2D and 3D shapes, vectors, dot product, cross product, and vector equations. Understand conic sections, lines, planes, spheres, and more.

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Analytical Geometry Essentials: 2D, 3D, Vectors, Projections

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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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