Dot Product & Cross Product of two vectors

Work done by a force F s θ θ W = F s cosθ = F · s F s

Dot product (Scalar product) a · b = |a| |b| cosθ = a x b x + a y b y + a z b z 0 o <θ<180 o is the angle between vectors a and b a · c = |a| |c| cos90 o = 0 a and c are perpendicular or orthogonal. a · d = |a| |d| cos 0 0 = |a| |d| a · a = |a| |a| cos 0 0 = |a| 2 θ b a c d

Properties of Dot Product Commutative property a ·b = b·a Distributive property a · ( b + c ) = a ·b + b·c

Example a = (1, 2, 4), b =(-1, 2, -1) a · b = 1x(-1) + 2x2 + 4x(-1) = -1

a = (0, 1, -1), b = (2, -1, 1) a · b = 0x2 + 1x(-1) +(-1)x1 = -2

j k i ·= ·= j ·= ·= ·= ·= k ij jk i ik (1,0,0) ·(1,0,0) =1 (0,1,0) ·(0,1,0) =1 (0,0,1) ·(0,0,1) =1 (1,0,0) ·(0,1,0) =0 (0,1,0) ·(0,0,1) =0 (1,0,0) ·(0,0,1) =0 k i j x y z 1 1 1

Example Find the angle between vectors a = (1, 1, -1) a nd b = (2, -1, 0) a · b = 1x2 + 1x(-1) +(-1)x0 = 1 cos θ = = =

A(2,1, 0), B(1, -1,1), C(0, 2, 1) are three points. Find the angles in the triangle ABC α β θ A B C

Example a = α + +2, b= +β -, c= - +γ Find the numbers α, β, γ which make the vectors a, b and c mutually perpendicular. ijkijkijk

Example a = + +2, b= + - Construct any vector perpendicular to a and b ijkijk

Direction Cosines y a a z x θzθz θxθx θyθy i j k

Example Find the direction cosines of the vector

Example Find the unit vector in the direction of the vector a =(3, 4, 1).

Direction Ratios of a straight line To determine the inclination of a straight line. Components of any vector s that is parallel to line. s =p q r,, Direction Ratios of a straight line L : Line L

Example (Two dimension) Find a set of direction ratios for the straight line y=2x+1.

Example Find the equation for a straight line which passes though point(1, 0, -1) and has a set of direction ratios of (1, 2, 2).

Components of a vector a=(a x, a y, a z ) k i j x y z a (a x, a y, a z )·(1, 0,0)=a x (a x, a y, a z )·(0, 1,0)=a y (a x, a y, a z )·(0, 0,1)=a z

Rotation of Axes in Two dimensions… θ x y X Y P(x, y), P(X, Y) = (cos(π/2+θ), sin(π/2+ θ) = (-sin θ, cos θ) = (cosθ, sinθ) X = (x, y)·(cosθ, sin θ) = xcos θ + ysin θ Y = (x, y)·(-sinθ, cos θ) = -xsin θ + ycos θ

Rotation of Axes in Three Dimension… k i j x y z a I J K Z X Y a=(x, y, z) = x i+y j+zk in Oxyz a = (?, ?, ?) in OXYZ O

Rotation of Axes in Three Dimension… k i j x y z I J K Z X Y In OXYZ, I=(1, 0, 0) In Oxyz, I = (l 1, m 1, n 1 ) O J=(0, 1, 0) K=(0, 0, 1) J = (l 2, m 2, n 2 ) K = (l 3, m 3, n 3 ) l1l1 m1m1 n1n1

Rotation of Axes in Three Dimension… k i j x y z I J K Z X Y In xyz, i=(1, 0, 0) In OXYZ, i= (l 1, l 2, l 3 ) O j=(0, 1, 0) k=(0, 0, 1) j = (m 1, m 2, m 3 ) k = (n 1, n 2, n 3 ) l1l1 l2l2 l3l3

Rotation of axes OxyzOXYZ i(1, 0, 0)(l 1, l 2, l 3 ) j(0, 1, 0)(m 1, m 2, m 3 ) k(0, 0, 1)(n 1, n 2, n 3 ) I(l 1, m 1, n 1 )(1, 0, 0) J(l 2, m 2, n 2 )(0, 1, 0) K(l 3, m 3, n 3 )(0, 0, 1)

Rotation of Axes in Three Dimension… k i j x y z I J K Z X Y In OXYZ, i= (l 1, l 2, l 3 ) O j = (m 1, m 2, m 3 ) k = (n 1, n 2, n 3 ) P(x, y, z) or P(X, Y, Z) r r = x i+y j+z k = x (l 1 I + l 2 J + l 3 K) +y (m 1 I + m 2 J + m 3 K) +z (n 1 I + n 2 J + n 3 K) = (x l 1 + ym 1 + zn 1 )I + (x l 2 + ym 2 + zn 2 )J +(x l 3 + ym 3 + zn 3 )K

Rotation of Axes in Three Dimension… r = x i+y j+z k = (x l 1 + ym 1 + zn 1 )I + (x l 2 + ym 2 + zn 2 )J +(x l 3 + ym 3 + zn 3 )K =X I+Y J+Z K

Rotation of Axes in Three Dimension…

Plane x y z r O n a P(x 0, y 0, z 0 ) Q(x, y, z) ( a - r )· n = 0 r · n = a · n -- Vector equation of a plane ax+by+cz=ax 0 +by 0 +cz 0 or a(x-x 0 ) + b(y-y 0 ) +c(z-z 0 ) =0 If the normal n=(a, b, c), then the equation for the plane can be written as: QP = a-r QP · n = 0

Rotation of Axes in 3 Dimensions x z y X’ Y’ Z’ i j k J I K ^ ^ ^ ^ ^ ^

Rotation of Axes in 3 Dimensions x z y i j k ^ ^ ^ I ^ l1l1 m1m1 I = (l 1, m 1, n 1 ) ^ n1n1

Rotation of Axes in 3 Dimensions x z y i j k ^ ^ ^ J ^ l2l2 m2m2 n2n2 J = (l 2, m 2, n 2 ) ^

Rotation of Axes in 3 Dimensions x z y i j k ^ ^ ^ K ^ l3l3 m3m3 n3n3 K = (l 3, m 3, n 3 ) ^

Rotation of Axes in 3 Dimensions x z y X’ Y’ Z’ i j k J I K ^ ^ ^ ^ ^ ^ l1l1 l2l2 l3l3 i = (l 1, l 2, l 3 ) In the X’, Y’, Z’ system ^

Rotation of Axes in 3 Dimensions x z y X’ Y’ Z’ i j k J I K ^ ^ ^ ^ ^ ^ j = (m 1, m 2, m 3 ) In the X’, Y’, Z’ system ^ m2m2 m1m1 m3m3

Rotation of Axes in 3 Dimensions x z y X’ Y’ Z’ i j k J I K ^ ^ ^ ^ ^ ^ k = (n 1, n 2, n 3 ) In the X’, Y’, Z’ system ^ n1n1 n2n2 n3n3

Rotation of Axes in 3 Dimensions x z y X’ Y’ Z’ i j k J I K ^ ^ ^ ^ ^ ^ P(x, y, z) or P(X’, Y’, Z’) are related by Direction Cosines

Example Find the equation of a line which passes through P(1, 2, -6) and is parallel to the vector (3, 1, -1)

Example Find the equation of a plane which passes through P(1, 2, -6) and is perpendicular to the vector (3, 1, - 1)