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Basics of Analytical Geometry

By

Kishore Kulkarni

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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 => 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 byax2 + 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

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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 < x2 + y2 + z2 < 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 < 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 + v2

2 + v32)

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Dot (Scalar) Product of vectors

Dot product of two vectors a = a1i + a2j + a3kand b = b1i + b2j + b3k is defined asa.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|)

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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 cos2 α + cos2 β + cos2 γ = 1. Vector a = |a| <cos α, cos β, cos γ>

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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| sinθ 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 asV = 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 byr = 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 byn.(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 m:n is given by p = (mb + na) / (m+n)which corresponds toP = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n), (mz2 + nz1)/(m+n))