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 ax 2 + by 2 + 2hxy + 2gx + 2fy + c = 0 where abc + 2fgh – af 2 – bg 2 – ch 2 = 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: x 2 + y 2 = r 2, r => radius of circle  Parabola: y 2 = 4ax or x 2 = 4ay  Ellipse: x 2 /a 2 + y 2 /b 2 =1, a is major axis & b is minor axis  Hyperbola: x 2 /a 2 - y 2 /b 2 =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 ax 2 + by 2 + 2hxy + 2gx + 2fy + c = 0 where abc + 2fgh – af 2 – bg 2 – ch 2 != 0  Special cases h 2 = ab, it is a parabola h 2 < ab, it is an ellipse h 2 > ab, it is a hyperbola h 2 < ab and a=b, it is a circle

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

8 3D Geometry  Question What region does this inequality represent in a 3D space ? 9 < x 2 + y 2 + z 2 < 25

9 3D Geometry  Straight Lines Parametric equations of line passing through (x 0, y 0, z 0 ) x = x 0 + at, y = y 0 + bt, z = z 0 + ct Symmetric form of line passing through (x 0, y 0, z 0 ) (x - x 0 )/a = (y - y 0 )/b = (z - z 0 )/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 and Q(a, b, c) in 3D space corresponds to position vector  Let P and Q then vector PQ = OQ – OP =  Length of a vector v = is given by |v| = sqrt(v v v 3 2 )

11 Dot (Scalar) Product of vectors  Dot product of two vectors a = a 1 i + a 2 j + a 3 k and b = b 1 i + b 2 j + b 3 k is defined as a.b = a 1 b 1 + a 2 b 2 + a 3 b 3.  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 = a 1 i + a 2 j + a 3 k 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 = a 1 / |a|, cos β = a 2 / |a|, cos γ = a 3 / |a|.  Hence cos 2 α + cos 2 β + cos 2 γ = 1.  Vector a = |a|

13 Cross (Vector) Product of vectors  Cross product of two vectors a = a 1 i + a 2 j + a 3 k and b = b 1 i + b 2 j + b 3 k is defined as a x b = (a 2 b 3 – a 3 b 2 )i +(a 3 b 1 – a 1 b 3 )j +(a 1 b 2 – a 2 b 1 )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(x 0, y 0, z 0 ) is given by r = r 0 + tv where r 0 =, r =, v = is a vector parallel to L, t is a scalar.  Equation of a plane is given by n.(r - r 0 ) = 0 where n is a normal vector, which is analogous to the scalar equation a (x- x 0 ) + b (y- y 0 ) + c (z- z 0 ) = 0

17 Vector Equations  Let a and b be position vectors of points A(x 1, y 1,z 1 ) and B(x 2, y 2,z 2 ). 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 = ((mx 2 + nx 1 )/(m+n), (my 2 + ny 1 )/(m+n), (mz 2 + nz 1 )/(m+n))