Jewett_PhysicsforSciAndEng 11e_PPT_ch02 2

[Audio] Physics for Scientists and Engineers, 11e Chapter 2: Vectors Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 1.

[Audio] 2.1 Coordinate System (1 of 2) Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.

[Audio] 2.1 Coordinate System (2 of 2)  = x r tan y x y r cos sin = = 2 2   r x y = + Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 3.

[Audio] Example 2.1: Polar Coordinates The Cartesian coordinates of a point in the xy plane are (x, y) = (−3.50, −2.50) m as shown in the figure. Find the polar coordinates of this point. 2 2 2 2 3.50 m 2.50 m 4.30 m r x y = + = − + − = ( ) ( ) = 215.5° Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 4.

[Audio] 2.2 Position, Distance, and Displacement (1 of 2) Table 2.1 Position of the Car at Various Times Position t (s) x (m) Ⓐ 0 30 Ⓑ 10 52 Ⓒ 20 38 Ⓓ 30 0 Ⓔ 40 −37 Ⓕ 50 −53 Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 5.

[Audio] 2.2 Position, Distance, and Displacement (2 of 2) Displacement x of a particle: change in position in a given time interval. It has both magnitude and direction (vector). f i x x x   − Distance: the length of a path followed by a particle. It only has a magnitude (no direction). Vector quantity requires specification of both direction and magnitude. Scalar quantity has numerical value and no direction. Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 6.

[Audio] Quick Quiz 2.1 Which of the following choices best describes what can be determined exactly from the table and figures in the earlier slide for the entire 50-s interval? (a) The distance the car moved (b) The displacement of the car (c) Both (a) and (b) (d) Neither (a) nor (b) (d) Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 7.

[Audio] September 2 Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 8.

[Audio] 2.3 Vector and Scalar Quantities A scalar quantity is completely specified by a single value with an appropriate unit and has no direction. A vector quantity is completely specified by a number with an appropriate unit (the magnitude of the vector) plus a direction. Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 9.

[Audio] Quick Quiz 2.2 Which of the following are vector quantities, and which are scalar quantities? (a) Your age (b) Position (c) Velocity (d) Speed (e) Mass Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 10.

[Audio] 2.4 Basic Vector Arithmetic (1 of 5) r r only if A B = = A B Both vectors point in the same direction along parallel lines. Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 11.

[Audio] 2.4 Basic Vector Arithmetic (2 of 5) Resultant vector (𝑹) ─ head to tail method: vector that completes the polygon and is drawn from the tail of the first vector to the tip of the last vector. Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 12.

[Audio] 2.4 Basic Vector Arithmetic (3 of 5) r r r r ( ) + = + A B B A commutative law of addition Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 13.

[Audio] 2.4 Basic Vector Arithmetic (4 of 5) r r r r r r ( ) ( ) ( ) A B C A B C associative law of addition + + = + + Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 14.

[Audio] 2.4 Basic Vector Arithmetic (5 of 5) Vector subtraction: r r r r r r ( ) A B A B − = + − ( ) 0 A + −A = r scalar multiplication: mA Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 15.

[Audio] Quick Quiz 2.3 A  B  r r r = + R A B The magnitudes of two vectors and are 𝐴 = 12 units and 𝐵 = 8 units. Which pair of numbers represents the largest and smallest possible values for the magnitude of the resultant vector ? (a) 14.4 units, 4 units (b) 12 units, 8 units (c) 20 units, 4 units (d) None of these answers (c) Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 16.

[Audio] Quick Quiz 2.4 B  A  If vector is added to vector , which two of the following choices must be true for the resultant vector to be equal to zero? a. and are parallel and in the same direction. A  B  b. and are parallel and in opposite directions. A  B  c. and have the same magnitude. B  A  d. and are perpendicular. A  B  (b) And (c) Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 17.

[Audio] Example 2.2: A Vacation Trip A car travels 20.0 km due north and then 35.0 km in a direction 60.0° west of north, as shown in the figure. Find (a) the magnitude and (b) the direction of the car's resultant displacement. 𝑅 = Ԧ𝐴 + 𝐵 2 2 2 cos R A B AB  = + − 𝑎) 𝑇ℎ𝑒 𝑐𝑜𝑠𝑖𝑛𝑒 𝑙𝑎𝑤: = 48.2 km b) 𝑇ℎ𝑒 𝑠𝑖𝑛𝑒 𝑙𝑎𝑤: sin 𝛽 ⟹ sin 𝛽 = 𝐵 𝑅 sin 𝜃 𝐵 = sin 𝜃 𝑅 = sin 𝛼 𝐴 𝛽 = 38.9° Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 18.

[Audio] 2.5 Components of a Vector and Unit Vectors (1 of 8) x 2 2 cos cos =  =   x A A A = + x y A y y sin sin   =  = 1 tan y   =   A A A A A A A A A  − x   Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 19.

[Audio] 2.5 Components of a Vector and Unit Vectors (2 of 8) Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 20.

[Audio] 2.5 Components of a Vector and Unit Vectors (3 of 8) Cartesian coordinate system ˆ i ˆ j ˆ k Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 21.

[Audio] 2.5 Components of a Vector and Unit Vectors (4 of 8) r ˆ ˆ x y A A A i j = + Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 22.

[Audio] 2.5 Components of a Vector and Unit Vectors (5 of 8) Polar coordinate system Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 23.

[Audio] 2.5 Components of a Vector and Unit Vectors (6 of 8) Vector Addition using Components r r r x y x y A A B B = + = + + + R A B i j i j ( ) ( ) ˆ ˆ ˆ ˆ r x x y y A B A B = + + + R i j ( ) ( ) ˆ ˆ r ˆ ˆ R i j R R = +  x y R A B = + x x x R A B = + y y y Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 24.

[Audio] 2.5 Components of a Vector and Unit Vectors (7 of 8) Magnitude of a Vector 2 2 2 2 R R R A B A B = + = + + + x y x x y y ( ) ( ) R A B y y y tan R A B  + = = + x x x Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 25.

[Audio] 2.5 Components of a Vector and Unit Vectors (8 of 8) Vectors in Three Dimensions r r ˆ ˆ ˆ A i j k A A A = + + x y z x x y y z z A B A B A B R i j k = + + + + + r ( ) ( ) ( ) ˆ ˆ ˆ ˆ ˆ ˆ B i j k B B B = + + x y z R A B = + z z z 2 2 2 R R R R = + + x y z R y cos x x z y R  = R cos R  = R  = R cos z Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 26.

[Audio] Quick Quiz 2.5 How does a component of a vector compare with the magnitude of the vector? (a) It is always larger than the magnitude of the vector. (b) It is never larger than the magnitude of the vector. (c) It is sometimes larger than the magnitude of the vector (b) Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 27.

[Audio] Quick Quiz 2.6 For which of the following vectors is the magnitude of the vector equal to one of the components of the vector?  a. ˆ A = 2 +5ˆ i j  b. = 3ˆ − B j  c. C = + 5ˆ k (c) Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 28.

[Audio] Example 2.3: The Sum of Two Vectors (1 of 2) Find the sum of two vectors and lying in the xy plane and given by A  B  r $ $ and A = (2.0i+2.0j) r $ $ B = (2.0i 4.0j) − Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 29.

[Audio] Example 2.4: The Resultant Displacement A particle undergoes three consecutive displacements: , and , and . ( ) 1r 15i 30j 12k cm ˆ ˆ ˆ  = + + ( ) 2r 23i ˆ 14j 5.0k cm ˆ ˆ  = − − ( ) 3r 13i ˆ 15j cm ˆ  = − + Find unit – vector notation for the resultant displacement and magnitude. r r r r r r r r  =  +  +  1 2 3 ˆ ˆ 15 23 13 cm 30 14 15 cm i j = + − + − + ( ) ( ) k ˆ 12 5.0 0 cm + − + ( ) 2 2 2 x y z R R R R = + + ˆ ˆ ˆ = 25 31 7.0 cm i j k + + ( ) 2 2 2 25 cm 31 cm 7.0 cm 40 cm = + + = ( ) ( ) ( ) Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 30.

[Audio] Example 2.5: Taking a Hike (1 of 3) A hiker begins a trip by first walking 25.0 km southeast from her car. She stops and sets up her tent for the night. On the second day, she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger's tower. (A) Determine the components of the hiker's displacement for each day. Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 31.

[Audio] Example 2.5: Taking a Hike (2 of 3) (B) Determine the components of the hiker's resultant displacement for the trip. Find and expression for in terms of unit vectors. R  R  C) After reaching the tower, the hiker wishes to return to her car along a single straight line. What are the components of the vector representing this hike? What should the direction of the hike be? Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 32.

[Audio] September 9 Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 33 33.

[Audio] 2.6 The Scalar Product of Two Vectors (1 of 3) Ԧ𝐴 ∙ 𝐵 ≡ 𝐴𝐵 cos 𝜃 Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 34.

[Audio] 2.6 The Scalar Product of Two Vectors (2 of 3) Commutative property: Ԧ𝐴 ∙ 𝐵 = 𝐵 ∙ Ԧ𝐴 Distributive property: Ԧ𝐴 ∙ 𝐵 + Ԧ𝐶 = Ԧ𝐴 ∙ 𝐵 + Ԧ𝐴 ∙ Ԧ𝐶 Special cases: Ԧ𝐴 ⊥ 𝐵 𝜃 = 90° ⟹ Ԧ𝐴 ∙ 𝐵 = 0 Ԧ𝐴 ∥ 𝐵 𝜃 = 0° ⟹ Ԧ𝐴 ∙ 𝐵 = 𝐴𝐵 Ԧ𝐴 ↑↓ 𝐵 𝜃 = 180° ⟹ Ԧ𝐴 ∙ 𝐵 = −𝐴𝐵 Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 35.

[Audio] 2.6 The Scalar Product of Two Vectors (3 of 3) Ƹ𝑖 ∙ Ƹ𝑗 = Ƹ𝑗 ∙ ෠𝑘 = ෠𝑘 ∙ Ƹ𝑖 = 0 1 ˆ ˆ · · ˆ ˆ ˆ · ˆ = = = i i j j k k r ˆ ˆ ˆ x y z A A A = + + A i j k r B i j k ˆ ˆ ˆ x y z B B B = + + r r ·B x x y y z z A A B A B A B = + + r r 2 2 2 2 · x y z A A A A A = + + = A Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 36.

[Audio] Quick Quiz 2.7 Which of the following is true about the relationship between the dot product of two vectors and the product of the magnitudes of the vectors? a. is larger than AB A B    b. is smaller than AB. A B r r g A B r r g c. could be larger or smaller than AB, depending on the angle between the vectors. d. could be equal to AB. A B r r g (d) Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 37.

[Audio] Example 2.6 The Scalar Product r The vectors and are given by and r . ˆ 2 ˆ B i j = − + B  A  3 ˆ ˆ 2 A i j = + (A)Determine the scalar product . A B r r g (B) Find the angle between Ԧ𝐴 and 𝐵. Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 38.

[Audio] 2.7 The Vector Product of Two Vectors (1 of 3) The cross product of vectors and : A r B r r r r C = A B  Its magnitude: C AB sin = The direction of Ԧ𝐶 is perpendicular to the plane formed by Ԧ𝐴 and 𝐵, it is determined by using the right-hand rule illustrated in the figure. Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 39.

[Audio] 2.7 The Vector Product of Two Vectors (2 of 3) r r r r Vector product is not Commutative: A × B = B × A − Distributive property: ( ) A × B + C = A × B + A × C r r r r r r r Ԧ𝐴 ⊥ 𝐵 𝜃 = 90° ⟹ Ԧ𝐴 × 𝐵 = 𝐴𝐵 Special cases: Ԧ𝐴 ∥ 𝐵 𝜃 = 0° ⟹ Ԧ𝐴 × 𝐵 = 0 Ԧ𝐴 ↑↓ 𝐵 𝜃 = 180° ⟹ Ԧ𝐴 × 𝐵 = 0 The derivative of the vector product with respect to some variable such as t is r r r r r r ( ) d dA dB dt dt dt  =  +  A B B A Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 40.

[Audio] 2.7 The Vector Product of Two Vectors (3 of 3) ˆ ˆ ˆ ˆ ˆ l j j l k  = −  = $ $ 0 ˆ ˆ ˆ ˆ j ı ı k k  = j  =  = ˆ ˆ ˆ ˆ ˆ j k k j i  = −  = ˆ ˆ ˆ ˆ ˆ k l l k j  = −  = $ ˆ ˆ j r r y z x y z x $ ˆ ˆ A B j  = = + + x y z y z x y z x ı k A A A A A A A A A ı k B B B B B B B B B x y z r r B ˆ ˆ A j A B A B ı A B A B A B A B k  = − + − + − y z z y z x x z x y y x ( ) ( )$ ( ) Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 41.

[Audio] Quick Quiz 2.8 Which of the following about the relationship between the magnitude of the cross product of two vectors and the product of the magnitudes of the vectors is true?   a. is larger than AB. A B   A B b. is smaller than AB.  A B   c. could be larger or smaller than AB, depending on the angle between the vectors.   A B d. could be equal to AB. (d) Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 42.

[Audio] Example 2.7 The Vector Product r r r The vectors and are given by and Find and verify . ˆ 2 ˆ B i j = − + A r r A B  B r 3 ˆ ˆ 2 A i j = + r r r r that . A B B A  = −  r r ( ) ( ) ( ) ( )ˆ 2 3 2 2 ˆ 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3ˆ 3 2 ˆ A B i j i j i i i j j i j j  = +  − + =  − +  +  − +  0 ˆ ˆ 4 ˆ 0 3 7 k k k = + + + = r r ( ) ( ) ( ) ( ) 3 ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 2 3 2 2 2 ˆ ˆ ˆ 2 ˆ 3ˆ B A i j i j i i i j i j j j  = − +  + = −  + −  +  +  0 ˆ ˆ 3 ˆ 0 4 7 k k k = − − + = −   Therefore, . A B B A  = −  Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 43.

[Audio] 2.6) Vector Ԧ𝐴 has a magnitude of 29 units and points in the positive y direction. When vector 𝐵 is added to Ԧ𝐴, the resultant vector Ԧ𝐴 + 𝐵 points in the negative y direction with a magnitude of 14 units. Find the magnitude and direction of 𝐵. Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 44.

[Audio] 2.9) The displacement vectors Ԧ𝐴 and 𝐵 shown in Figure P2.9 both have magnitudes of 3 m. The direction of vector Ԧ𝐴 is 𝜃 = 30°. Find graphically a) Ԧ𝐴 + 𝐵 b) Ԧ𝐴 − 𝐵 c) 𝐵 − Ԧ𝐴 d) Ԧ𝐴 − 2𝐵 Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 45.

[Audio] 2.24) Vector 𝐵 has x, y, and z components of 4, 6, and 3 units, respectively. Calculate a) the magnitude of 𝐵 and b) the angle that 𝐵 makes with each coordinate axis. Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 46.

[Audio] 2.36) Using the definition of the scalar product, find the angles between Ԧ𝐴 = −2 Ƹ𝑖 − 4 Ƹ𝑗 and 𝐵 = 3 Ƹ𝑖 − 4 Ƹ𝑗 + 2෠𝑘. Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 47.

[Audio] 2.37) Given 𝑀 = 2 Ƹ𝑖 − 3 Ƹ𝑗 + ෠𝑘 and 𝑁 = 4 Ƹ𝑖 + 5 Ƹ𝑗 − 2෠𝑘 calculate the vector product 𝑀 × 𝑁. Ƹ𝑖 + 8 Ƹ𝑗 + 22෠𝑘 Serway | Jewett | Peroomian, Physics for Scientists and Engineers with Modern Physics, 11th Edition. © 2026 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 48.