# Assignment 5 - Assignment 5 1 We will start by drawing a...

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Chapter 14 / Exercise 27
Elementary and Intermediate Algebra
Tussy/Gustafson
Expert Verified
Assignment 5 1. We will start by drawing a diagram of the two vectors, tail to tail, with an angle of 55 ° Using the parallelogram law of vector addition, completing the parallelogram and drawing the diagonal that represents the sum of ´ u and ´ v Finding Size of all Angles θ = 360 2 ( 55 ) 2 = 125 ° Calculating the length of the diagonal, using the cosine law, | u + v | 2 = | u | 2 + | v | 2 2 | u | v cosθ ¿ 3 2 + 5 2 2 ( 3 ) ( 5 ) cos 125 ° ¿ 51.207
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Chapter 14 / Exercise 27
Elementary and Intermediate Algebra
Tussy/Gustafson
Expert Verified
| u + v | = 51.207 ¿ 7.155 7.2 The magnitude of the resultant is approximately 7.2 units. To find the angle, we will use the sine law sin ( A ) a = sin ( B ) b = sin ( C ) c sinθ | ´ u | = sin125 ° | ´ u + ´ v | sinθ = 3sin 125 ° 7.2 sinθ = 0.3413 θ≈ 20 ° The magnitude of the resultant is approximately 7.2 units and it makes an angle 20 ° with ´ v .
2. a.
b. 3. a. ´ p q = 6 ( 2,1 ) 4 ( 1, 3 ) ¿ ( 6 × 2,6 × 1 ) + ( 4 × 1, 4 × 3 ) ¿ ( 12,6 ) + ( 4,12 ) ¿ ( 12 +(− 4 ) , 6 + 12 ) ¿ (− 16,18 ) For a point with the Cartesian coordinates (a,b) the magnitude of the resultant vector from the origin would be, ´ | u | = a 2 + b 2
¿ ( 16 ) 2 + 18 2 ¿ 580 ¿ 24.08 24.1 units Angle θ made with the Cartesian Axes, θ = tan 1 b a ¿ tan 1 ( 18 16 ) ¿ tan 1 ( 1.125 ) ¿ 48.36 °≈ 48.4 ° Since this point is in the Second Quadrant of the Cartesian system, the angle made in the standard form would be 180 48.4 ° = 131.6 ° Hence the coordinates of the point of the vector (-16,18) and the magnitude is 24.1 units. b. ´ u v = 3 ( 11,1, 2 ) 5 ( 2,4,6 ) ¿ ( 3 × 11,3 × 1,3 × 2 ) +(− 5 × 2, 5 × 4, 5 × 6 ) ¿ ( 33,3, 6 ) + ( 10, 20, 30 ) ¿ ( 33 + ( 10 ) , 3 + ( 20 ) , 6 + ( 30 ) ) ¿ ( 23, 17, 36 ) For a point with the Cartesian coordinates (a,b,c) the magnitude of the resultant vector from the origin would be, | ´ u | = a 2 + b 2 + c 2 Hence , | ´ u v | = 23 2 + ( 17 ) 2 + ( 36 ) 2
¿ 2114 ¿ 45.97 46 units Hence the coordinates of the vector are (23,-17,-36) and magnitude is 46 units. 4. a. 3 ´ x 4 ( 2 ´ x + ´ y ) + 10 ´ y ¿ 3 ´ x 8 ´ x 4 ´ y + 10 ´ y ¿ 5 ´ x + 6 ´ y b. 9 ( 2 ´ x −´ y + 3 ´ x ) 2 ( 6 ´ x 3 ´ y ) ¿ 18 ´ x 9 ´ y + 27 ´ x 12 ´ x + 6 ´ y ¿ 33 ´ x 3 ´ y c. 2 ( 4 ´ x −´ y )+ 10 ( 2 ´ x 3 ´ y )− 7 x 3 ´ y ) ¿ 8 ´ x 2 ´ y + 20 ´ x 30 ´ y 7 ´ x + 21 ´ y ¿ 21 ´ x 11 ´ y 5. The point P(x,y) represents the vector in Cartesian form. The horizontal component is the length of x and the vertical component is the length of y.
The primary trigonometric ratios of this right-angled triangle: ´ ¿ u ¿ cosθ = x ¿ ´ ¿ u ¿ sinθ = y ¿ To find the horizontal component (the length of x), we use the cosine ratio. We know the angle we are working with is 75°. We also know the length of the hypotenuse is ¿ ´ u ¿ 8 x = ¿ ´ u | cosθ ¿ 8 × cos75 ° ¿ 8 × 0.258 ¿ 2.07 2.1 To find the horizontal component (the length of y), we use the sine ratio. y = | ´ u | sinθ ¿ 8 × sin 75 ° ¿ 8 × 0.96 ¿ 7.72 7.7 Therefore, the vector in Cartesian form is ´ u =( 2.1,7.7 ) .