Calculus 3 Sections 12.1 and 12.2 Reading Assignment: 3D Coordinates and Vectors

Answer Only Exercise 2 by using a screenshot provided Calculus Pearson textbook. Make sure you read these three questions very carefully and see on what it is asking for and what is really about. Please be very serious careful with this assignment of exercise #2.

References: Thomas' Calculus: Early Transcendentals | Calculus | Calculus | Mathematics | Store | Pearson+

Exercise 2. Read the subsection "Vector Algebra Operations" (p. 721 -722). Explain the geometric process for adding vectors described here. Do not explain the algebraic formula, but describe how this is done visually in terms of the graph.Chapter 12 Vectors and Geometry of Space 12.2 Vectors 721 EXAMPLE 1 Find the (a) component form and (b) length of the vector with initial point P(-3. 4, 1) and terminal point Q(-5, 2, 2). Solution (a) The standard position vector v representing PO has components 1=1 - 1 =-5- (-3) =-2, why = 12 - 1 =2-4= -2, and 1 -3-2-2-1-1. The component form of PO is v = (-2, -2, 1 ). (b) The length or magnitude of v = PQ is F - (a, b) [v/ = V(-2) + (-2) + (1) = V9 = 3. EXAMPLE 2 A small cart is being pulled along a smooth horizontal floor with a FIGURE 12.11 The force pulling the 20-lb force F making a 45" angle to the floor (Figure 12.11). What is the effective force cart forward is represented by the vector F moving the cart forward? whose horizontal component is the effect tive force (Example 2). Solution The effective force is the horizontal component of F = (a. b) , given by a = [F| cos 450 = (20) V2 2 = 14.14 1b.Notice that F is a two-dimensional vector. Vector Algebra Operations Two principal operations involving vectors are vector addition and scalar multiplication. A scalar is simply a real number, and is called such when we want to draw attention to the differences between numbers and vectors, Scalars can be positive, negative, or zero and are used to "scale" a vector by multiplication. (a) DEFINITIONS Let u = (#1. #2. my ) and v = ( v. v2, v; ) be vectors with & a scalar Addition: utv = (u, + v , My + up, My + vy) Scalar multiplication: ku - (kaj- kuz, kus) We add vectors by adding the corresponding components of the vectors. We multiply a vector by a scalar by multiplying each component by the scalar. The definitions also apply to planar vectors, except in that case there are only two components, (#1, w2 ) and The definition of vector addition is illustrated geometrically for planar vectors in Fig- (b) ure 12.12a, where the initial point of one vector is placed at the terminal point of the other. FIGURE 12.12 (a) Geometric inter- Another interpretation is shown in Figure 12.12b (called the parallelogram law of addi- pretation of the vector sum. (b) The tion), where the sum, called the resultant vector, is the diagonal of the parallelogram. In parallelogram law of vector addition physics, forces add vectorially as do velocities, accelerations, and so on. So the force act- in which both vectors are in standard ing on a particle subject to two gravitational forces, for example, is obtained by adding the position. two force vectors.Chapter 12 Vectors and Geometry of Space 722 Chapter 12 Vectors and the Geometry of Space Figure 12.13 displays a geometric interpretation of the product ku of the scalar & and vector u. If & > 0. then ku has the same direction as u; if &