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Scalars and Vectors in Physics

Learn about the differences between scalars and vectors, vector addition and subtraction, resolving vectors into components, and vector products. Includes examples and explanations.

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Scalars and Vectors in Physics

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  1. Vectors • Scalars: a physical quantity described by a single number • Vector: a physical quantity which has a magnitude (size) anddirection. • Examples: velocity, acceleration, force, displacement. • A vector quantity is indicated by bold face and/or an arrow. • The magnitude of a vector is the “length” or size (in appropriate units). • The magnitude of a vector is always positive. • The negative of a vector is a vector of the same magnitude put opposite direction (i.e. antiparallel)

  2. Combining scalars and vectors • scalars and vectors cannot be added or subtracted. • the product of a vector by a scalar is a vector • x = ca x = |c| a (note combination of units) • if c is positive, x is parallel to a • if c is negative, x is antiparallel to a

  3. Vector addition • most easily visualized in terms of displacements • Let X = A + B+ C • graphical addition: place A and B tip to tail; X is drawn from the tail of the first to the tip of the last • A + B = B + A B A X X A B

  4. B B A + = = A R C C D B B C A B D A D A • Vector Addition: Graphical Method of R = A + B • Shift B parallel to itself until its tail is at the head of A, retaining its original length and direction. • Draw R (the resultant) from the tail of A to the head of B. the order of addition of several vectors does not matter

  5. -B R B -B A A - = + = A • Vector Subtraction: the negative of a vector points in the opposite direction, but retains its size (magnitude) • A-B = A +( -B)

  6. A Ay Ax A Ay q Ax • Resolving a Vector (2-d) • replacing a vector with two or more (mutually perpendicular) vectors => components • directions of components determined by coordinates or geometry. A= Ax + Ay Ax = x-component Ay = y-component q Be careful in 3rd , 4th quadrants when using inverse trig functions to find q. Component directions do not have to be horizontal-vertical!

  7. Vector Addition by components • R = A + B + C • Resolve vectors into components(Ax, Ay etc. ) • Add like components • Ax + Bx + Cx = Rx • Ay + By + Cy = Ry • The magnitude and direction of the resultant R can be determined from its components. • in general R ¹ A + B + C • Example 1-7: Add the three displacements: • 72.4 m, 32.0° east of north • 57.3 m, 36.0° south of west • 72.4 m, straight south

  8. Unit Vectors • a unit vector is a vector with magnitude equal to 1 (unit-less and hence dimensionless) • in the Cartesian coordinates: Right Hand Rule for relative directions: thumb, pointer, middle for i, j, k. Express any vector in terms of its components:

  9. B A f B cosf • Products of vectors (how to multiply a vector by a vector) • Scalar Product (aka the Dot Product) • is the angle between the vectors • A.B = Ax Bx +Ay By +Az Bz = B.A • = B cos f A is the portion of B along A times the magnitude of A • = A cos f B is the portion of A along B times the magnitude of B • note: the dot product between perpendicular vectors is zero.

  10. Example: Determine the scalar product between • A = (4.00m, 53.0°) and B = (5.00m, 130.0°)

  11. Products of vectors (how to multiply a vector by a vector) • Vector Product (aka the Cross Product) 3-D always! • is the angle between the vectors • Right hand rule: A´B = C • A – thumb • B – pointer • C – middle • Cartesian Unit vectors • C = AB sin f • = B sinf A is the portion of B perpendicular A times the magnitude of A • = A sinf B is the portion of A perpendicular B times the magnitude of B

  12. C = AB sin f • = B sin f A is the part of B perpendicular A times A • = A sin f B is the part of A perpendicular B times B B sinf B A f Write vectors in terms of components to calculate cross product

  13. Example: A is along the x-axis with a magnitude of 6.00 units, B is in the x-y plane, 30° from the x-axis with a magnitude of 4.00 units. Calculate the cross product of the two vectors.

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