Dot Product & Cross Product of two vectors

Work done by a forceFsW = F s cos = F s

Fs

Dot product (Scalar product)a b = |a| |b| cos = axbx + ayby + azbz

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Properties of Dot ProductCommutative propertya b = ba

Distributive propertya ( b + c ) = a b + bc

Examplea = (1, 2, 4), b =(-1, 2, -1)

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

Examplea = (0, 1, -1), b = (2, -1, 1)

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

Example======(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 xyz111

ExampleFind the angle between vectors a = (1, 1, -1) and b = (2, -1, 0)

a b = 1x2 + 1x(-1) +(-1)x0 = 1

cos = =

=

ExampleA(2,1, 0), B(1, -1,1), C(0, 2, 1) are three points. Find the angles in the triangle ABCABC

Examplea = + +2 , b= + - , c= - +Find the numbers , , which make the vectors a, b and c mutually perpendicular.

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

Direction Cosinesyazxzxy

ExampleFind the direction cosines of the vector

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

Direction Ratios of a straight lineTo 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.

ExampleFind 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=(ax, ay, az)xyz111a(ax, ay, az)(1, 0,0)=ax(ax, ay, az)(0, 1,0)=ay(ax, ay, az)(0, 0,1)=az

Rotation of Axes in Two dimensionsxyXYP(x, y), P(X, Y)= (cos(/2+), sin(/2+ )= (-sin , cos )= (cos, sin)X= (x, y)(cos, sin )= xcos + ysin Y= (x, y)(-sin, cos )= -xsin + ycos

Rotation of Axes in Three DimensionxyzaIJKZXYa=(x, y, z) = x i+y j+zk in Oxyza = (?, ?, ?) in OXYZO

Rotation of Axes in Three DimensionxyzIJKZXYIn OXYZ,I=(1, 0, 0)

In Oxyz,I = (l1, m1, n1)

OJ=(0, 1, 0)K=(0, 0, 1)

J = (l2, m2, n2) K = (l3, m3, n3)

l1m1n1

Rotation of Axes in Three DimensionxyzIJKZXYIn xyz,i=(1, 0, 0)

In OXYZ,i= (l1, l2, l3)

Oj=(0, 1, 0)k=(0, 0, 1)

j = (m1, m2, m3) k = (n1, n2, n3)

l1l2l3

Rotation of axes

Rotation of Axes in Three DimensionxyzIJKZXYIn OXYZ, i= (l1, l2, l3)

Oj = (m1, m2, m3) k = (n1, n2, n3)

P(x, y, z) or P(X, Y, Z)rr = x i+y j+z k = x (l1I + l2J + l3K) +y (m1I + m2J + m3K) +z (n1I + n2J + n3K)= (x l1+ ym1+ zn1)I + (x l2+ ym2+ zn2)J +(x l3+ ym3+ zn3)K

Rotation of Axes in Three Dimensionr = x i+y j+z k = (x l1+ ym1+ zn1)I + (x l2+ ym2+ zn2)J +(x l3+ ym3+ zn3)K=X I+Y J+Z K

Rotation of Axes in Three Dimension

PlanexyzrOnaP(x0 , y0 , z0)Q(x, y, z)( a - r ) n = 0r n = a n-- Vector equation of a planeax+by+cz=ax0+by0+cz0

or a(x-x0) + b(y-y0) +c(z-z0) =0If the normal n=(a, b, c), then the equation for the plane can be written as:

Rotation of Axes in 3 Dimensions

Rotation of Axes in 3 Dimensions

Rotation of Axes in 3 Dimensions

Rotation of Axes in 3 Dimensions

Rotation of Axes in 3 Dimensions

Rotation of Axes in 3 Dimensions

Rotation of Axes in 3 Dimensions

Rotation of Axes in 3 DimensionsP(x, y, z) or P(X, Y, Z)are related by Direction Cosines

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

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