LEC Scalar and Vector PDF

Scalar and Vector Mark Anthony C. Burgonio, MSc Assistant Professional Lecturer, College of Science, Department of Physical Sciences Polytechnic University of the Philippines  Outline Scalar vs Vector Vector Representations Un...

Scalar and Vector Mark Anthony C. Burgonio, MSc Assistant Professional Lecturer, College of Science, Department of Physical Sciences Polytechnic University of the Philippines  Outline Scalar vs Vector Vector Representations Unit Vectors Vector Addition  Scalar and Vector Scalar Quantities Vector Quantities - quantities that can be - quantities that can be completely completely described by a described by both a magnitude magnitude (or by a number and (“how much or how big”) and a units) alone. direction in space. Examples: Examples: o speed, 30 m⁄s o displacement, 20 m from the origin o mass, 50 kg o velocity, 60 m⁄s South of East o temperature, 37 ℃ o force, 50 N to the right Representation of a Vector Quantity 𝑦 Analytic Magnitude-direction form: 𝐴 𝑎$ 𝐴⃗ = 𝐴𝑎$ Magnitude of a vector 𝐴⃗ : 𝐴 = 𝐴⃗ 𝑨 Based on coordinate 𝜃 system used: (component 𝑥 form) 𝐴⃗ = 𝐴! + 𝐴" + 𝐴#  Components of a Vector y x-component of 𝐀: A A% ϕ A$ = Acos θ (wrt x-axis) θ A$ = Asin ϕ (wrt y-axis) A$ x y-component of 𝐀: Analogy: A% = Asin θ (wrt x-axis) Magnitude of A → hypotenuse A A% = Acos ϕ (wrt y-axis) A$ → adjacent to θ; opposite to ϕ A% → opposite to θ; adjacent to ϕ  Component of a Vector (+) 𝑦 Remarks: The component of a vector may be II I positive or negative depending on which quadrant the vector lies. (+) 𝜃 Quadrant x-component y-component 𝑥 I + + (−) II − + III − − III IV IV + − (−)  Example #1 Determine the x and y components of each of the forces shown below. F& = 80 N, 40° above the +x-axis F' = 120 N, 70° above the +x-axis F( = 150 N, 35° above the -x-axis Magnitude and Direction of a Vector y A = A = Magnitude of A A= A'$ + A'% A A% ϕ For angle with respect to x −axis: )& A% θ θ = tan A$ A$ x For angle with respect to y −axis: )& A$ Given the components, how do we calculate ϕ = tan A% the magnitude and direction of a vector?  Bearing Form of a Vector Descriptions of Direction: N North of East (N of E) East of North (E of N) North of West (N of W) West of North (W of N) A South of West (S of W) A% West of South (W of S) ϕ East of South (E of S) South of East (S of E) θ W E Example: if 𝐴 = 10.0 N and 𝜃 = 30.0°, A$ S 𝑨 = 10.0 N, 30.0° N of E, or 𝑨 = 10.0 N, 60.0° E of N  Determining the Correct Direction Angle Sign of x- Sign of y- Always start with a vector component component Angle diagram before doing any Measured calculations. + + counterclockwise from the +x-axis Examine the signs of the Measured clockwise components of the vector and − + from the –x-axis. which quadrant the vector lies Measured Use trigonometric relations on − − counterclockwise from –x-axis right triangles. Identify the Measured clockwise correct angle + − from +x-axis  Example #2 Vector 𝐴⃗ has y-component 𝐴" = +6.0m. 𝐴⃗ makes an angle of 32.0° counterclockwise ⃗ (b) What is the magnitude of 𝐴? from the +y-axis. (a) What is the x-component of 𝐴? ⃗ A" Solutions: (a) The x-component of A is opposite to θ. 32° A! A A! tan θ = A" A! = A" tan θ = (6.0m)tan(32.0°) A! = 3.75m A! = −3.8m (b) The magnitude of A is given by: A= A#! + A#" A= 3.8m # + (6.0m)# = 7.1m  2D Vector and Its Unit Vector y The unit vector or normalized vector of a vector is defined as: (recall, A = A@a) a@ A a@ ≡ A A where: A% A= A'$ + A'% A The unit vector has a magnitude of 1 and x indicates only the direction of a vector. A$  Unit Vectors in Cartesian Plane y If we divide the perpendicular component of a vector by its magnitude, we will obtain either + 1 or −1. For the x-coordinate, the unit vector is ı.̂ +ı̂ if towards the +x-axis −ı̂ if towards the −x-axis D̂ For the y-coordinate, the unit vector is ȷ.̂ +ȷ̂ if towards the +y-axis Ĉ x −ȷ̂ if towards the −y-axis  Decomposition of a 2D Vector 𝑦 When decomposing a vector, unit vectors provide a useful way to write component 𝑨 vectors: A $ = A $ ı̂ 𝐴$ A % = A % ȷ̂ ,̂ 𝐴# The full decomposition of the vector A can then be written: +̂ 𝑥 A = A $ + A % = A $ ı̂ + A % ȷ ̂  Decomposition of a 3D Vector 𝑧 In 3D rectangular coordinate system: ;̂ - unit vector along the x-axis. 𝑨