Complex numbers #2 - Q&A

Question 1 A bird flies in the xy-plane with a position vector given by 𝑟 ⃗ = ( 𝛼 𝑡 − 𝛽 𝑡 3 ) 𝑖 ^ + 𝛾 𝑡 2 𝑗 ^ r =(αt−βt 3 ) i ^ +γt 2 j ^ ​ with 𝛼 = 2.4   𝑚 𝑠 ,   𝛽 = 1.6   𝑚 𝑠 3   and   𝛾 = 4.0   𝑚 𝑠 2 . α=2.4 s m ​ ,β=1.6 s 3 m ​ andγ=4.0 s 2 m ​ . The positive y-direction is vertically upward. At 𝑡 = 0 t=0 the bird is at the origin. Calculate the velocity vector of the bird as a function of time. Calculate the acceleration vector of the bird as a function of time. What is the bird's altitude (y-coordinate) as it flies over x=0 for the first time after? Answer 1 A bird flies in the xy-plane with a position vector given by 𝑟 ⃗ = ( 𝛼 𝑡 − 𝛽 𝑡 3 ) 𝑖 ^ + 𝛾 𝑡 2 𝑗 ^ , r =(αt−βt 3 ) i ^ +γt 2 j ^ ​ , with 𝛼 = 2.4   𝑚 𝑠 ,   𝛽 = 1.6   𝑚 𝑠 3   and   𝛾 = 4.0   𝑚 𝑠 2 . α=2.4 s m ​ ,β=1.6 s 3 m ​ andγ=4.0 s 2 m ​ . The positive y-direction is vertically upward. At 𝑡 = 0 t=0 the bird is at the origin. The velocity vector of the bird as a function of time ( 𝑣 = 𝑑 𝑟 ⃗ / 𝑑 𝑡 v=d r /dt) is: 𝑣 ⃗ = ( 2.4 − 4.8 𝑡 2 ) 𝑖 ^ + 8.0 𝑡 𝑗 ^ v =(2.4−4.8t 2 ) i ^ +8.0t j ^ ​ The acceleration vector of the bird as a function of time ( 𝑎 = 𝑑 𝑣 ⃗ / 𝑑 𝑡 a=d v /dt) is: 𝑎 ⃗ = ( − 9.6 𝑡 ) 𝑖 ^ + 8.0 𝑗 ^ a =(−9.6t) i ^ +8.0 j ^ ​ Answer 2 The velocity vector is 𝑣 ( 𝑡 ) = ( 𝛼 − 𝛽 𝑡 2 ) 𝑖 ^ + ( 𝛾 𝑡 ) 𝑗 ^ v(t)=(α−βt 2 ) i ^ +(γt) j ^ ​ The position vector can be found by integrating the velocity vector. 𝑣 ( 𝑡 ) = ∫ 𝑣 ( 𝑡 ) 𝑑 𝑡 = ( 𝛼 𝑡 − 𝛽 𝑡 3 3 + 𝐶 1 ) 𝑖 ^ + ( 𝛾 𝑡 2 2 + 𝐶 2 ) 𝑗 ^ v(t)=∫v(t)dt=(αt− 3 βt 3 ​ +C 1 ​ ) i ^ +( 2 γt 2 ​ +C 2 ​ ) j ^ ​ At 𝑡 = 0 t=0, the bird is at the origin, so the integration constants can be determined. 𝑟 ( 0 ) = 𝐶 1 𝑖 ^ + 𝐶 2 𝑗 ^ = 0 r(0)=C 1 ​ i ^ +C 2 ​ j ^ ​ =0 Therefore, 𝐶 1 C 1 ​ and 𝐶 2 C 2 ​ are equal to zero. 𝑟 ( 𝑡 ) = ( 𝛼 𝑡 − 𝛽 𝑡 3 3 ) 𝑖 ^ + ( 𝛾 𝑡 2 2 ) 𝑗 ^ r(t)=(αt− 3 βt 3 ​ ) i ^ +( 2 γt 2 ​ ) j ^ ​ The acceleration vector is the derivative of the velocity vector with respect to time. 𝑎 ( 𝑡 ) = 𝑑 𝑣 𝑑 𝑡 = − 2 𝛽 𝑡 𝑖 ^ + 𝛾 𝑗 ^ a(t)= dt dv ​ =−2βt i ^ +γ j ^ ​ We will first find the time when the x-component of the position vector is equal to zero. 𝛼 𝑡 = 𝛽 𝑡 3 3 αt= 3 βt 3 ​ 𝑡 2 = 3 𝛼 𝛽 t 2 = β 3α ​ 𝑡 = 3 × 2.4 1.6 = 2.12 𝑠 t= 1.6 3×2.4 ​ ​ =2.12s We will plug this value into the y-component of the position vector. 𝑦 ( 𝑡 ) = 𝛾 𝑡 2 2 y(t)= 2 γt 2 ​ 𝑦 ( 2.12 ) = 4 ( 2.12 ) 2 2 = 9 𝑚 y(2.12)= 2 4(2.12) 2 ​ =9m Question 2 Use De Moivre’s Theorem to find the indicated power of the complex number. Write your answer in standard form 𝑎 + 𝑏 𝑖 a+bi and support with a calculator. ( 1 − 3 𝑖 ) 3 (1− 3 ​ i) 3 Answer 1 𝑎 + 𝑏 𝑖 = 1 − 3 𝑖 a+bi=1− 3 ​ i Convert to polar form. ( 𝑎 , 𝑏 ) = ( 1 , − 3 ) (a,b)=(1,− 3 ​ ) is in quadrant 4, so the angle must also agree. 𝑟 = 𝑎 2 + 𝑏 2 = 1 2 + ( − 3 ) 2 = 4 = 2 r= a 2 +b 2 ​ = 1 2 +(− 3 ​ ) 2 ​ = 4 ​ =2 𝜃 = arctan ⁡ 𝑏 𝑎 = arctan ⁡ − 3 1 = − 𝜋 3 θ=arctan a b ​ =arctan 1 − 3 ​ ​ =− 3 π ​ 1 − 3 𝑖 = 𝑟 ( cos ⁡ 𝜃 + 𝑖 sin ⁡ 𝜃 ) 1− 3 ​ i=r(cosθ+isinθ) = 2 ( cos ⁡ ( − 𝜋 3 ) + 𝑖 sin ⁡ ( − 𝜋 3 ) ) =2(cos(− 3 π ​ )+isin(− 3 π ​ )) Use De Moivre’s Question 3 Rhode Island is the smallest state in the United States. Its land area is about 2.9 × 1 0 10 2.9×10 10 square feet. Alaska, the largest state, is about 5.5 × 1 0 2 5.5×10 2 times as large as Rhode Island. What is the land area of Alaska in square feet? Answer 1 When multiplying two numbers in scientific notation, you should group the significands (the fancy word for the number in front of the power of ten) and the powers of ten. Hence 2.9 × 1 0 10 × 5.5 × 1 0 2 = ( 2.9 × 5.5 ) × ( 1 0 10 × 1 0 2 ) = 15.95 × 1 0 12 = 1.595 × 1 0 13 2.9×10 10 ×5.5×10 2 =(2.9×5.5)×(10 10 ×10 2 )=15.95×10 12 =1.595×10 13 At this point, we should look at how many significant figures our starting numbers had. They both had 2 in this case. So our result should too. So we need to round so that this number has only two digits. = 1.6 × 1 0 13 =1.6×10 13 Question 4 Multiplicative Inverse of a Complex Number The multiplicative inverse of a complex number 𝑧 z is a complex number 𝑧 𝑚 zm such that 𝑧 × 𝑧 𝑚 = 1 z×zm=1. Find the multiplicative inverse of complex number. 𝑧 = 1 + 𝑖 z=1+i Answer 1 The multiplicative inverse of complex number 𝑧 = 𝑧 𝑚 z=zm such that 𝑍 × 𝑧 𝑚 = 1 Z×zm=1. Hence, the multiplicative inverse of a complex number can be given as 𝑍 𝑚 = 1 𝑍 Z m ​ = Z 1 ​ . Given: 𝑍 = 1 + 𝑖 Z=1+i 𝑍 𝑚 = 1 1 + 𝑖 ⋅ 1 − 𝑖 1 − 𝑖 = 1 − 𝑖 ( 1 + 𝑖 ) ( 1 − 𝑖 ) = 1 − 𝑖 1 + 1 = 1 − 𝑖 2 = 1 2 − 1 2 𝑖 Z m ​ = 1+i 1 ​ ⋅ 1−i 1−i ​ = (1+i)(1−i) 1−i ​ = 1+1 1−i ​ = 2 1−i ​ = 2 1 ​ − 2 1 ​ i Question 5 What’s the square root of − 10 −10? Answer 1 Step 1 The square root of − 10 −10 is a complex number, which is expressed as 𝑖 10 i 10 ​ This is because 𝑖 2 = − 1 i 2 =−1 and − 10 = − 1 × 10 −10=−1×10 so we can express − 1 −1 ​ as 𝑖 i Answer 2 Step 1 In the domain of real numbers, this would be invalid, but in the domain of complex numbers − 10 = 𝑖 10 = 3.16227766017 𝑖 −10 ​ =i 10 ​ =3.16227766017i Answer 3 Step 1 Square root of − 10 −10 is a complex number. Complex numbers consist of real numbers and an imaginary number denoted as 𝑖 i. We say it is imaginary because we use it to solve the impossible equation 𝑖 2 = − 1 i 2 =−1. Therefore, to solve for the square root of a negative number, you multiply the square root of − 1 −1 (denote by 𝑖 i) by the square root of the positive (or absolute) value of the negative number. Hence, − 10 = 𝑖 × 10 −10 ​ =i× 10