Basics of Analytical Geometry

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

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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

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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

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Conic Sections

• Circle x2 y2 r2 , r gt 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)

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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 lt ab, it is an ellipse
• h2 gt ab, it is a hyperbola
• h2 lt ab and ab, it is a circle

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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)

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3D Geometry

• Question
• What region does this inequality represent in a
  3D space ?

9 lt x2 y2 z2 lt 25
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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.

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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
  lt a, bgt and Q(a, b, c) in 3D space corresponds to
  position vector lt a, b, cgt
• Let P ltx1, y1, z1gt and Q lt x2, y2, z2 gt then
  vector PQ OQ OP lt x2 x1, y2 y1, z2
  z1gt
• Length of a vector v lt v1, v2, v3gt is given by
• v sqrt(v12 v22 v32)

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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 ? is the angle between a and b, we can write
• a.b abcos?
• Hence a.b 0 implies two vectors are orthogonal.
  
• Further a.b gt 0 we can say that they are in the
  same general direction and a.b lt 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)

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Direction Angles and Direction Cosines

• Direction Angles a, ß, ? 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 a a1/ a, cos ß a2/ a, cos ? a3/
  a.
• Hence cos2 a cos2 ß cos2 ? 1.
• Vector a a ltcos a, cos ß, cos ?gt

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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 sin? represents area of
  parallelogram.

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Cross (Vector) Product

• Question
• What can you say about the cross product of
  two vectors in 2D ?

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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.
  

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Vector Equations

• Equation of a line L with a point P(x0, y0, z0)
  is given by
• r r0 tv
• where r0 lt x0, y0, z0gt, r lt x, y, zgt, v
  lta, b, cgt 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

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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 mn is given by
• p (mb na) / (mn)
• which corresponds to
• P ((mx2 nx1)/(mn), (my2 ny1)/(mn), (mz2
  nz1)/(mn))