Question

These are Questions/answers for #1, 2, 3, 17, 18, 19 & 20 if you can't see what it says in the picture (Please answer all questions in correct order)
#1.Given w = −96i −180j, what are the magnitude and direction of −4w? Round the answers to the nearest whole number.

204; 62°

816; 62°

204; 242°

816; 242°


# 2. A basketball player shoots a free throw, where the position of the ball is modeled by x = (24cos 48°)t and y = 6.1 + (24sin 48°)t − 16t 2. What is the height of the ball, in feet, when it is 13 feet from the free throw line? Round to three decimal places.

10.053

10.292

10.673

11.025


#3.Vectors u and v are shown on the graph.

vectors u and v share an initial point and vector u points down 8 units and vector v points to the right 6 units

Which of the following vectors represents u + v? (Please put/show a graph if you can)


17.What are the magnitude and direction of u + v + w? Round the magnitude to three decimal places and the direction to the nearest degree.

53.241; 355°

52.822; 355°

53.241; 5°

52.822; 5°


#18. Which point represents z1 + z2?

P

Q

R

S


Question 19. A barge is being towed by two tugboats, using ropes with force vectors as shown.

Two vectors named F sub 1 and F sub 2 that share an initial point from the barge each pointing to a different tugboat

Given vector F sub 1 equals open angled bracket 12000 comma 7000 close angled bracket and vector F sub 2 equals open angled bracket 14500 comma negative 5000 close angled bracket comma what is the angle between the ropes? Round to the nearest degree.

31°

33°

49°

54°


Question 20.Given v = 8i − 3j and w = −12i + 4j, find 7v − 2w.

−80i − 29j

−80i + 29j

80i − 29j

80i + 29j

Answer 1

Answer:

See below for answers and explanations (along with a graph for #3)

Step-by-step explanation:

Problem #1

Applying scalar multiplication, $-4w=-4\langle-96,-180\rangle=\langle384,720\rangle$.

Its magnitude would be $||-4w||=\sqrt{384^2+720^2}=816$.

Its direction would be $\displaystyle\theta=tan^{-1}\biggr(\frac{720}{384}\biggr)\approx61.927^\circ\approx62^\circ$.

Thus, B) 816; 62° is the correct answer

Problem #2

Find the time it takes for the ball to cover 13ft:

$x=(24\cos48^\circ)t\\13=(24\cos48^\circ)t\\t\approx0.8095$

Find the height of the ball at the time it takes for the ball to cover 13ft:

$y=6.1+(24\sin48^\circ)t-16t^2\\y=6.1+(24\sin48^\circ)(0.8095)-16(0.8095)^2\\y\approx10.053$

Thus, A) 10.053 is the correct answer

Problem #3

We have $u=\langle0,-8\rangle$ and $v=\langle6,0\rangle$ as our vectors. Thus, $u+v=\langle0+6,-8+0\rangle=\langle6,-8\rangle$. Attached below is the correct graph. You can also solve the problem visually by using the parallelogram method where the resultant vector is the diagonal of the parallelogram.

Problem 4 (#7)

$\displaystyle t \cdot v=(7)(-10)+(-3)(-8)=-70+24=-46$

Thus, C) -46 is the correct answer

Problem 5 (#8)

Find $r$ and $\theta$:

$r=\sqrt{x^2+y^2}=\sqrt{2^2+(-8)^2}=\sqrt{4+64}=\sqrt{68}=2\sqrt{17}\approx8.246$

$\displaystyle\theta=tan^{-1}\biggr(\frac{y}{x}\biggr)=tan^{-1}\biggr(\frac{-8}{2}\biggr)\approx-75.964^\circ$

Find the true direction angle accounting for Quadrant IV:

$\theta=360^\circ-75.964^\circ=284.036^\circ$

Write the complex number in polar/trigonometric form:

$z=8.246(\cos284.036^\circ+i\sin284.036^\circ)$

Thus, C) 8.246(cos 284.036° + i sin 284.036°) is the correct answer

Problem 6 (#12)

Eliminate the parameter and find the rectangular equation:

$x=3t\\\frac{x}{3}=t\\ \\y=t^2+5\\y=(\frac{x}{3})^2+5\\y=\frac{x^2}{9}+5\\9y=x^2+45\\0=x^2-9y+45\\x^2-9y+45=0$

Thus, D) x^2-9y+45=0 is the correct answer

Problem 7 (#13)

Find the magnitude of the vector:

$||v||=\sqrt{(-77)^2+36^2}=85$

Find the true direction of the vector accounting for Quadrant II:

$\displaystyle \theta=tan^{-1}\biggr(\frac{36}{-77}\biggr)\approx-25^\circ=180^\circ-25^\circ=155^\circ$

Write the vector in trigonometric form:

$w=85\cos155^\circ i+85\sin155^\circ j$

Thus, D) w=85cos155°i+85sin155°j is the corrwect answer

Problem 8 (#15)

$\frac{z_1+z_2}{2}=\frac{(3-7i)+(-9-19i)}{2}=\frac{-6-26i}{2}=-3-13i=(-3,-13)$

Thus, C) (-3,-13) is the correct answer

Problem 9 (#16)

Treat the golf ball and wind as vectors:

$u=\langle1.3\cos140^\circ,1.3\sin140^\circ\rangle$ <-- Golf Ball

$v=\langle1.2\cos50^\circ,1.2\sin50^\circ\rangle$ <-- Wind

Add the vectors:

$u+v=\langle1.3\cos140^\circ+1.2\cos50^\circ,1.3\sin140^\circ+1.2\sin50^\circ\rangle\approx\langle-0.225,1.755\rangle$

Find the magnitude of the resultant vector:

$||u+v||=\sqrt{(-0.225)^2+1.755^2}\approx1.769$

Find the true direction of the resultant vector accounting for Quadrant II:

$\displaystyle \theta=\tan^{-1}\biggr(\frac{1.755}{-0.225}\biggr)\approx-82.694^\circ\approx-83^\circ=180^\circ-83^\circ=97^\circ$

Thus, B) 1.769 m/s; 97° is the correct answer

Problem 10 (#17)

Identify the vectors and add them:

$u+v+w=\langle50\cos20^\circ,50\sin20^\circ\rangle+\langle13\cos90^\circ,13\sin90^\circ\rangle+\langle35\cos280^\circ,35\sin280^\circ\rangle=\langle50\cos20^\circ+13\cos90^\circ+35\cos280^\circ,50\sin20^\circ+13\sin90^\circ+35\sin280^\circ\rangle=\langle53.062,-4.367\rangle$

Find the magnitude of the resultant vector:

$||u+v+w||=\sqrt{53.062^2+(-4.367)^2}\approx53.241$

Find the true direction of the resultant vector accounting for Quadrant IV:

$\displaystyle \theta=\tan^{-1}\biggr(\frac{-4.367}{53.062}\biggr)\approx-4.705^\circ\approx-5^\circ=360^\circ-5^\circ=355^\circ$

Thus, A) 53.241, 355° is the correct answer

Problem 11 (#18)

We observe that $z_1=-8-6i$ and $z_2=4-4i$, hence, $z_1+z_2=(-8-6i)+(4-4i)=-4-10i$

Thus, Q is the correct answer

Problem 12 (#19)

Find the dot product of the vectors:

$F_1\cdot F_2=(12000*14500)+(7000*-5000)=174000000+(-35000000)=139000000$

Find the magnitude of each vector:

$||F_1||=\sqrt{12000^2+7000^2}=1000\sqrt{193}\\||F_2||=\sqrt{14500^2+(-5000)^2}=500\sqrt{941}$

Find the angle between the two vectors:

$\displaystyle \theta=\cos^{-1}\biggr(\frac{F_1\cdot F_2}{||F_1||||F_2||}\biggr)\\ \theta=\cos^{-1}\biggr(\frac{139000000}{(1000\sqrt{193})(500\sqrt{941})}\biggr)\\\theta\approx49.282^\circ\approx49^\circ$

Thus, C) 49° is the correct answer

Problem 13 (#20)

Using scalar multiplication, $7v-2w=7\langle8,-3\rangle-2\langle-12,4\rangle=\langle56,-21\rangle-\langle-24,8\rangle=\langle56-(-24),-21-8\rangle=\langle80,-29\rangle=80i-29j$

Thus, A) -80i - 29j is the correct answer