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Dot Product & Cross Product of two vectors

Dot Product & Cross Product of two vectors. Work done by a force. F. W = F s cos θ. θ. = F · s. s. F. θ. s. Dot product (Scalar product). c. b. a · b = |a| |b| cos θ = 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

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Dot Product & Cross Product of two vectors

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  1. Dot Product & Cross Product of two vectors

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

  3. Dot product (Scalar product) c b • a · b = |a| |b| cosθ = axbx + ayby + azbz • 0o <θ<180o is the angle between vectors a and b • a · c = |a| |c| cos90o = 0 a and c are perpendicular or orthogonal. • a · d = |a| |d| cos 00 = |a| |d| • a · a = |a| |a| cos 00 = |a|2 θ d a

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

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

  6. Example • a = (0, 1, -1), b = (2, -1, 1) a · b = 0x2 + 1x(-1) +(-1)x1 = -2

  7. k k k k k j j j j j i i i i i Example (1,0,0) ·(1,0,0) =1 · = · = (0,1,0) ·(0,1,0) =1 z · = (0,0,1) ·(0,0,1) =1 1 · = (1,0,0) ·(0,1,0) =0 1 · = (0,1,0) ·(0,0,1) =0 1 y · = (1,0,0) ·(0,0,1) =0 x

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

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

  10. k k k j j j i i i Example • a=α + +2 , b= +β - , c= - +γ • Find the numbers α, β, γ which make the vectors a, b and cmutually perpendicular.

  11. k k j j i i Example • a= + +2 , b= + - • Construct any vector perpendicular to a and b

  12. k a j i Direction Cosines z a θz θy θx y x

  13. Example • Find the direction cosines of the vector

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

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

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

  17. 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).

  18. k j i Components of a vector a=(ax, ay, az) (ax, ay, az)·(1, 0,0)=ax z (ax, ay, az)·(0, 1,0)=ay a 1 (ax, ay, az)·(0, 0,1)=az 1 1 y x

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

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

  21. k j i Rotation of Axes in Three Dimension… Z z In OXYZ, I=(1, 0, 0) J=(0, 1, 0) K=(0, 0, 1) K J Y n1 y In Oxyz, I =(l1, m1, n1) O m1 l1 I J = (l2, m2, n2) K = (l3, m3, n3) x X

  22. k j i Rotation of Axes in Three Dimension… Z z In xyz, i=(1, 0, 0) j=(0, 1, 0) k=(0, 0, 1) K J Y l3 y In OXYZ, i=(l1, l2, l3) O l2 I l1 j = (m1, m2, m3) k = (n1, n2, n3) x X

  23. Rotation of axes

  24. k j i Rotation of Axes in Three Dimension… Z P(x, y, z) or P(X, Y, Z) In OXYZ, i= (l1, l2, l3) z j = (m1, m2, m3) k = (n1, n2, n3) r K J Y y O r = x i+y j+z k = x (l1I + l2J + l3K) +y(m1I + m2J + m3K) +z(n1I + n2J + n3K) I = (x l1+ ym1+ zn1)I + (x l2+ ym2+ zn2)J +(x l3+ ym3+ zn3)K x X

  25. Rotation of Axes in Three Dimension… r = x i+y j+z k = (xl1+ ym1+ zn1)I + (xl2+ ym2+ zn2)J +(xl3+ ym3+ zn3)K =X I+Y J+Z K

  26. Rotation of Axes in Three Dimension…

  27. QP = a-r QP · n = 0 Plane z Q(x, y, z) P(x0 , y0 , z0) n ( a - r )· n = 0 r a r · n = a · n y O -- Vector equation of a plane If the normal n=(a, b, c), then the equation for the plane can be written as: x ax+by+cz=ax0+by0+cz0 or a(x-x0) + b(y-y0) +c(z-z0) =0

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

  29. z ^ I = (l1, m1, n1) n1 y ^ k ^ m1 j ^ I l1 ^ i x Rotation of Axes in 3 Dimensions

  30. z ^ n2 J = (l2, m2, n2) m2 y ^ J ^ k ^ j l2 ^ i x Rotation of Axes in 3 Dimensions

  31. z ^ K = (l3, m3, n3) n3 m3 y ^ k ^ j ^ K l3 ^ i x Rotation of Axes in 3 Dimensions

  32. z ^ i = (l1, l2, l3) In the X’, Y’, Z’ system Y’ y l2 Z’ ^ J ^ k ^ j X’ ^ K ^ I l1 l3 ^ i x Rotation of Axes in 3 Dimensions

  33. z ^ j = (m1, m2, m3) In the X’, Y’, Z’ system Y’ m2 y Z’ m3 ^ J ^ k ^ m1 j X’ ^ K ^ I ^ i x Rotation of Axes in 3 Dimensions

  34. z ^ k = (n1, n2, n3) In the X’, Y’, Z’ system Y’ n2 n3 y Z’ ^ J ^ k ^ j n1 X’ ^ K ^ I ^ i x Rotation of Axes in 3 Dimensions

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

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

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

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