THREE DIMENSIONAL GEOMETRY. skew lines

Size: px

Start display at page:

Download "THREE DIMENSIONAL GEOMETRY. skew lines"

Arleen Stafford
6 years ago
Views:

1 Class XII - Math Unit: s and Three Dimensial Geometry Ccepts and Formulae VECTORS Positio n Directi cosines Directi ratios Additio n Definiti Definiti Relati betwee n drs dcs and magnit ude of the vector Laws The positi vector of point P (x,y,z ) with respect to the origin is given by: OP = r = x + y + z If the positi vector OP of a point P makes angles α, β and γ with x, y and z axis respectively, then α, β and γ are called the directi angles and cosα, cosβ and cosγ are called the Directi cosines of the positi vector. The magnitude (r), directi ratios (a, b, c) and directi cosines ( l, m, n) of any vector are related as: a m c l=, m=, n= r r r Triangle Law: Suppose two vectors are represented by two sides of a triangle in sequence, then the third closing side of the triangle represents the sum of the two vectors PQ + QR = PR Parallelogram Law: If two vectors a and b are represented by two adjacent sides of a parallelogram in magnitude and directi, then their sum a +b is represented in magnitude and directi by the diagal of the parallelogram. OA??+ OB = OC

2 Prcpert ies of vector a`ditio n Multipli catiof of a vec 0or by a scalar joining two points Compo nent Form Commu tative ropert y Associa tive propert y Definiti Properti es Definiti Magnitu de Operati s For any two vectors a and b, a + b = b + a For any three vectors a,b and c, ( a + b) + c = a + ( b + c) If a is a vector and λ a scalar. Product of vector a by the scalar λis λa. Also, λa = λ a Let a and b be any two vectors and k and m being two scalars then (i)ka +ma =(k+m) a (ii)k(ma )= (km) a (iii)k(a +b )=ka +kb The vector PP joining points P (x, y, z ) and P (x, y, z ) (O is the origin) is given by: PP = OP OP The magnitude of vector PP is given by PP = (x x ) + (y y ) + (z z ) r = x i+ y j + z k in compent form Equality of vectors a = a î + a ĵ+ a3 k b = b î + b ĵ+ b3 k a = b a = b, a = b and a 3 = b 3 a a = î a + ĵ a + 3 k and b b = î b + ĵ b + 3 k

3 Additi of vectors a + b a =( + ) î a +( b + ) ĵ a +( 3 b + 3 ) k Subtracti of vectors a - b a b =( - ) î a +( b - ) ĵ a +( 3 - b 3 ) k a and b are collinear b = λ a. where λ is a n zero scalar. Product of Two s Scalar (or dot) product of two vectors Properti es of scalar Product Projecti of a vector Scalar product of two nzero vectors a and b, denoted by a.b= a b cos θ,where θ is the angle a and b, 0 (i)a b is a real number. (ii)if a and b are n zero vectors then a b =0 a b. (iii) Scalar product is commutative :a b =b.a (iv)if θ =0 then a b= a.b (v) If θ =π then a b=- a. b (vi) scalar product distributeover additi Let a, b and cbethree vectors, then a (b+c)= a b + a c (vii)let a and b be two vectors, and λ be any scalar. Then ( λa).b=( λa).b= λ(a.b)=a.( λb) (viii) Angle two n zero vectors a and b is given by cos θ = a.b a. b Projecti of a vector a other vector b is given by ˆ b a.b or a. or ( a.b ) b b

4 Secti formula Inequali ties (or cross) product of two vectors Properti es of cross product of vectors The positi vector of a point R dividing a line segment join P and Q whose positi vectors are a and b respectively, in na+mb (i) internally, is given by m + n mb-na (ii) externally, is given by m n Cauchy-Schwartz Inequality a.b a. b Triangle Inequality: The vector product of two nzero vectors a and b, denoted by a band defined as a b= a b sinθnˆ where, θ is the angle a and b,0 θ π and ˆn is a unit vector perpendicular to both a and b such thata,b and ˆn form a right handed system. (i) a b is a vector (ii) If a and b are n zero vectors then a b =0 iff a and b are collinear. π (iii) If θ =, then a b = a. b (iv) vector product distribute over additi If a,b andc are three vectors and λis a scalar, then (i) a (b+c)= ( a b ) + ( a c) (ii) λ (a b)=( λ a) b=a ( λb) (v) If we have two vectors a and b given in compent form as a=a ˆi+a ˆj+a kˆ and b=b ˆi+b ˆj+b kˆ 3 3 ˆi ˆj kˆ then a b= a a a 3 b b b 3

5 THREE DIMENSIONAL GEOMETRY Directi Cosines Definiti The directi cosines of the line joining P( x,y,z ) and Q( x,y,z ) are x -x y -y z z,, PQ PQ PQ where PQ= (x -x ) +(y -y ) +(z -z ) Skew Lines Definiti Skew lines are lines in space which are neither parallel nor intersecting. They lie in different s. Angle skew lines Angle skew lines is the angle two intersecting lines drawn from any point (preferably through the origin) parallel to each of the skew lines. Angle two lines The angle θ two vectors OA = a i + b j + c k and OB = a i + b j + c k is given b cos θ = a a +b b +c c a +b +c a +b +c a line Equati Cartesian Equati line passing through two given points equati of a line that passes through the given point whose positi vector is a and parallel to a given vector b is r=a+ λb Directi ratios of the line L are a, b, c. Then, cartesian form of equati of the line L is: x-x y-y z-z = = ) Equati The vector equati of a line which passes through two points whose positi vectors are a and b is r=a+ λ(b-a) ) Cartesian Equati

6 Cartesian equati of a line that passes through tw points (x, y, z ) and (x, y, z ) is x-x y-y z-z = = x -x y -y z -z Cditi for perpendicu larity Cditi for parallel lines Two lines with directi ratios a, a, a 3 and b, b, b 3 respectively are perpendicular if: a b + a b c c 0 Two lines with directi ratios a, a, a 3 and b, a b, b 3 respectively are parallel if a = b b = c c Shortest Distance two lines in space Distance two skew lines: ) form: Shortest distance two skew lines L and m, r = a + λb and r = a + µb is b b.(a - a ) d= b b ) Cartesian form The equatis of the lines in Cartesian form x-x a = y-y b = z-z x-x y-y z-z and = = c Then the shortest distance them is x -x y -y z -z d= (b c -b c ) +(c a -c a ) +(a b -a b ) Distance parallel lines Distance parallel lines b (a -a ) r = a + λb and r = a + µb is d= b In the vector form, equati of a which is at a distance d from the origin, and ˆn is the unit vector normal to the through the origin is

7 r.n ˆ = d a which is at a distance of d from the origin and the directi cosines of the normal to the as l, m, n is lx + my + nz = d. a perpendicular to a given line with directi ratios A, B, C and passing through a given point (x, y, z ) is A (x x ) + B (y y ) + C (z z ) = 0 a passing through three n collinear points (x, y, z ), (x, y, z ) and (x 3, y 3, z 3 ) is x-x y-y z-z x x y y z z x x y y z z =0 Intercept form of equati of. a passing through the intersectio n of two given s. Coplanarity of two lines a that makes intercepts a, b and c with x, y and z-axes respectively is x a + y b + z c = Any passing thru the intersecti of two s r. n =d and r. n =d is given by, r. n + λn = d + λd ( ) ) form: The given lines r = a + λb and r = a + µb are coplanar if and ly a a. b b = ( ) ( ) 0 ) Cartesian Form Let (x,y,z ) and (x,y,z ) be the coordinates of the points M and N respectively. Let a, b, c and a, b, c be the directi ratios

8 Angle two s form of b and respectively. The given lines are coplanar if and ly if x -x y -y z -z =0 If n and n are normals to the s r.n =d and r.n = d and θ is the angle the normals drawn from some comm point. n.n cos θ= n n Angle a line and a Distance of a point from a Cartesian form Let θ is the angle two s A x+b y+c z+d =0, A x+b y+c z+d =0 The directi ratios of the normal to the s are. cos θ = OP = r = x + y + z Let the angle the line and the normal to the = θ cosθ= b.n b n Distance of point P with positi vector a from a r.n =d is a.n-d where N is the normal to N the

For more information visit here:

For more information visit here: The length or the magnitude of the vector = (a, b, c) is defined by w = a 2 +b 2 +c 2 A vector may be divided by its own length to convert it into a unit vector, i.e.? = u / u. (The vectors have been denoted