LECTURE OUTLINE CHAPTER 3 Vectors in Physics
3-1 Scalars Versus Vectors Scalar a numerical value (number with units). May be positive or negative. Examples: temperature, speed, height, and mass. Vector a quantity with both magnitude, direction and unit. Examples: displacement and force.
3-1 Scalars Versus Vectors How to get to the library? need to know how far and which way.
3-2 The Components of a Vector Any vector A lying in xy plane can be resolved into two components, one in the x-direction and the other in the y-direction as shown in the figure.
3-2 The Components of a Vector The two components are: A = A cosθ x A = A sin θ y The magnitude of vector A = A + A θ = tan 2 2 x y 1 A A y x A
3-2 The Components of a Vector A vector is defined by its magnitude and direction, regardless of its location.
3-2 The Components of a Vector
3-2 The Components of a Vector Examples of vectors with components of different signs.
3-2 The Components of a Vector
3-2 The Components of a Vector
3-2 The Components of a Vector Sometimes we are given the angle between the vector and the y axis, as in the figure. If we call this angle θ, then: and A = A sin θ x A = A cosθ y
3-2 The Components of a Vector
3-3 Adding and Subtracting Vectors Adding vectors graphically: Place the tail of the second at the head of the first. The sum points from the tail of the first to the head of the last. The result vector is: R = A + B
3-3 Adding and Subtracting Vectors
3-3 Adding and Subtracting Vectors Vectors addition obeys the commutative law. C = A + B = B + A
3-3 Adding and Subtracting Vectors Adding Vectors Using Components: 1. Find the components of each vector to be added. 2. Add the x- and y-components separately. 3. Find the resultant vector.
3-3 Adding and Subtracting Vectors Example: A B Find : C x = 5.00m = 4.00m, C,, y C θ
3-3 Adding and Subtracting Vectors Answer : Adding component by component yields the components of vector C = Cx + C y
3-3 Adding and Subtracting Vectors
3-3 Adding and Subtracting Vectors Subtracting vectors. A B = A + ( B ) Where the negative vector is represented by an arrow of the same length as the original vector but pointing in the opposite direction.
3-3 Adding and Subtracting Vectors Subtracting Vectors: The negative of a vector is a vector of the same magnitude pointing in the opposite direction. Here, D= A B = A + ( B)
3-4 Unit Vectors Unit vectors are vectors that has a magnitude of one, and describe the directions in space. xˆ The x unit vector, is a dimensionless vector of unit length aligned with x-axis. The y unit vector ŷ, is a dimensionless vector of unit length aligned with y-axis.
3-4 Unit Vectors For any vector like A it can be written as A = A xˆ + A yˆ x y
3-4 Unit Vectors For any vector A, it can be written as A = aa a aa Where a is a unit vector in the direction of A A a = where a = 1 A
Multiplying vector by scalar. Multiplying vector by scalar. Multiplying vector by 3 increase its magnitude by a factor of 3, but does not change its direction. A = A xˆ + A yˆ x 3A = 3A xˆ + 3A yˆ x y y
Vectors Product 1. Scalar product or dot product ; the result of the product is a scalar quantity. A ib = AB cosθ A = A + A 2 2 x y B = B + B 2 2 x y
Vectors Product If θ = 90 so cosθ = 0 A ib = AB cosθ = 0 if A = A ˆ ˆ x x + A y y B = B xˆ + B yˆ A ib = A B + A B x y x x y y
Vectors Product so cosθ = A ib = A B + A B x x y y AB cos 1 A ib AB = θ
Vectors Product Vector product or cross product ; the result is a vector perpendicular to the plane of the two vectors. A B = AB sinθ ( A B ) = B A ( )
Vectors Product But A B = xˆ yˆ zˆ A x A y A B B B x y z z ( A B A B ) xˆ ( A B A B ) yˆ = y z z y x z z x + ( A B A B ) zˆ x y y x