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VECTORS

VECTORS. Vector: a quantity that is fully described by both magnitude (number and units) and direction. Scalar: a quantity that is described fully by magnitude alone. A vector is graphically represented by an arrow whose length reflects the magnitude and whose head reflects the direction.

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VECTORS

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  1. VECTORS Vector: a quantity that is fully described by both magnitude (number and units) and direction. Scalar: a quantity that is described fully by magnitude alone. A vector is graphically represented by an arrow whose length reflects the magnitude and whose head reflects the direction. In writing vector equations, vectors are represented by bold faced letters: vector: a + b = c scalar: a + b = c

  2. Notice Vector Direction: In relation to + x axis

  3. Vector Direction: By agreement, vectors are generally described by how many degrees the vector is rotated from the + x axis 30˚ 30˚ 150˚ Negative 2D vectors: A - A 180˚ opposite

  4. Vectors add trigonometrically, but also follow the commutative and associative laws of algebra: a + b = b + a a + (b + c) = (a + b) + c a - b = a + (-b) b -b A single vector can be resolved into right angle components. These components are traditionally resolved in terms of the x, y and z coordinate plane: z y x

  5. Resolving vectors in 2 dimensions: ax = a cos  ay = a sin  a ay  is always the angle the vector lies off of the +x axis! (from 0˚ to 360˚)  ax The components can (and will) specify the vector: a = √ ax2 + ay2 tan  = ay / ax

  6. When resolving a vector, it is conventional to describe the components in terms of unit length with the symbols i, j, and k representing unit vector lengths in the x, y and z directions. • in three dimensions: a = axi + ayj + azk • in two dimensions: a = 17 m @ 135˚ ax = acos135 = -12 m 135˚ ay = asin135 = 12 m 17m a = -12 m î + 12 m ĵ

  7. Adding Vectors A vector quantity can be properly expressed (unless otherwise specified) as a magnitude and direction, or as a sum of components. An automobile travels east for 32 km and then heads due south for 47 km. What is the magnitude and direction of its resultant displacement? s = 32km i - 47km j Ø = tan-1 (sy/sx) s = √ sx2 + sy2 = tan-1 (- 47/32) = -56˚ s = √ 322 + (-47)2 s = 57 km @ -56˚ or 304˚ = 57 km

  8. A woman leaves her house and walks east for 34 m. She then turns 25˚ to the south and walks for 46 m. At that point she head due west for 112 m. What is her total displacement relative to her house? s1 = s1xi + s1yj = 34 m i + 0 j s2 = s2xi + s2yj = 46cos(-25˚)i + 46sin(-25˚)j = 42m i - 19m j s3 = s3xi + s3yj = -112m i + 0j s = sxi + syj

  9. sx = s1x + s2x + s3x = (34 + 42 - 112)m = - 36m sy = s1y + s2y + s3y = (0 - 19 + 0)m = - 19 m s = - 36m i - 19m j If specifically asked for magnitude and direction: s = √ (- 36)2 + (-19)2 = 41 m Ø = tan-1 (sy / sx) s = 41 m @ 208˚ = 208˚ = - 19 - 36

  10. Subtracting Vectors  - Ĉ =  + (-Ĉ) and – Ĉ has the same magnitude as Ĉ but in the exact opposite direction: Ĉ Ĉ = Cxî + Cy ĵ - Ĉ - Ĉ = -Cxî - Cy ĵ

  11. Multiplication of Vectors 1) Multiplication of a vector by a scalar: multiply vector a by scalar c and the result is a new vector with magnitude ac, and in the same direction as a. (Ex: F = ma) 2) Multiplication of a vector by a vector to produce a scalar (called the dot product): a•b = abcos , where  is the angle between the vectors Work: W = Fr = Fcos•r

  12. 3) Multiplication of two vectors to produce a third vector (the cross product): a X b = c where the magnitude of c is defined by c = absin , where  is the angle between the vectors • the direction of a X b would be determined by the right hand rule (this will be explained in more depth later) • note that a X b would have the same magnitude as b X a, but be in the exact opposite direction! Ex: Torque: T = r x F

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