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.
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Transcript 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.
Basics of Analytical Geometry
By
Kishore Kulkarni
1
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
2
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
3
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
4
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 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
6
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)
7
3D Geometry
Question
What region does this inequality represent in a 3D
space ?
9 < x2 + y2 + z2 < 25
8
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.
9
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)
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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 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>
12
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 ?
14
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 = < 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
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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 to
P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n), (mz2 + nz1)/(m+n))
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