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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 Lines, Planes, Sphere, Cylinders
- Vectors
- 2D 3D Position Vectors
- Dot Product, Cross Product Box Product
- Analogy between Scalar and vector representations

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

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

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)

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

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)

3D Geometry

- Question
- What region does this inequality represent in a

3D space ?

9 lt x2 y2 z2 lt 25

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.

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)

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)

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

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.

Cross (Vector) Product

- Question
- What can you say about the cross product of

two vectors in 2D ?

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.

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

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))

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