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

Answer Only Exercise 3 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 #3.

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

Exercise 3. Read the first two paragraphs of the subsection "Unit Vectors" (p. 723). State what the standard unit vectors are and explain how to convert a vector in component form into a linear combination of the standard unit vectors.' Pay careful attention to the word "standard" for this exercise. Importantly, as vectors, these standard unit vectors should also have vector arrows on them. Unfortunately, in many parts of this material, there are multiple ways of writing the same thing. For this class, we will stick to these two ways of writing vectors, though this issue does pop up elsewhere too.Chapter 12 Vectors and Geometry of Space 12.2 Vectors 723 op, = xi+ yl+ zak Unit Vectors A vector v of length I is called a unit vector. The standard unit vectors are i = (1,0,0). j = (0, 1, 0), and k = (0, 0, 1). Any vector v = ( v. v2, v) ) can be written as a linear combination of the standard unit vectors as follows: PP V= ( v. 12. v) = (v1. 0,0) + (0. 12.0) + (0. 0. v,) = UI ( 1, 0, 0) + 22 (0, 1. 0) + v3 ( 0, 0, 1 ) = uit uj + uk. We call the scalar (or number) up the i-component of the vector v, u, the FIGURE 12.15 The vector from A to j-component, and v; the k-component. As shown in Figure 12.15, the component form Ris RR = (x - x)it (2 - yit for the vector from A(x, ), 2) to P(xz, 12, 22) is (2 - 3)k. AR = (12 - x)i + (2 - >j + (2 - z)k. If v * 0, then its length | v| is not zero and That is. v/ w is a unit vector in the direction of v, called the direction of the nonzero vector v. EXAMPLE 4 Find a unit vector u in the direction of the vector from A(1, 0. 1) to P,(3, 2, 0).Solution We divide PP, by its length: AR = (3 - 1)i + (2 - 0)j + (0 - 1)k = 2i + 2j - k [RAI = V(2) + (2) + (-1) = V4 + 4 + 1 = 19 =3 PP, 2i + 2j - k U = = WIN 3 it ;i - k. This unit vector u is the direction of PP- EXAMPLE 5 If v - 3i - 4j is a velocity vector, express v as a product of its speed times its direction of motion. Solution Speed is the magnitude (length) of v: |v = V(3) +(-4) = V9 + 16 =5. HISTORICAL BIOGRAPHY The unit vector v/ v| is the direction of v: Hermann Grassmann 3i - 4j (1809-1877) V lvl i 5 www. goo . gl/geHsw0 So v = 31 - 4) = i Length Direction of motion (speed)