1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Snowcat [4.5K]
2 years ago
10

Find the three angles of the triangle formed using the position vectors 2i hat − j + 5k and i hat + 2j + 4k and the line segment

connecting their endpoints. Give your answers in degrees to two decimal places.
Mathematics
1 answer:
victus00 [196]2 years ago
8 0

Given

\vec{a}=2\vec{i}-\vec{j}+5\vec{k},

\vec{b}=\vec{i}+2\vec{j}+4\vec{k},

you can find

\vec{a}-\vec{b}=2\vec{i}-\vec{j}+5\vec{k}-(\vec{i}+2\vec{j}+4\vec{k})=\vec{i}-3\vec{j}+\vec{k}.

Three vectors \vec{a}, \vec{b}, \vec{a}-\vec{b} form a triangle.

1.

\cos\angle 1=\dfrac{\vec{a}\cdot \vec{b}}{|\vec{a}|\cdot |\vec{b}|}=\dfrac{2\cdot 1+(-1)\cdot 2+5\cdot 4}{\sqrt{2^2+(-1)^2+5^2}\cdot \sqrt{1^2+2^2+4^2} }=\dfrac{20}{\sqrt{30} \cdot \sqrt{21} }.

2.

\cos\angle 2=\dfrac{\vec{a}\cdot (\vec{a}-\vec{b})}{|\vec{a}|\cdot |\vec{a}-\vec{b}|}=\dfrac{2\cdot 1+(-1)\cdot (-3)+5\cdot 1}{\sqrt{2^2+(-1)^2+5^2}\cdot \sqrt{1^2+(-3)^2+1^2} }=\dfrac{10}{\sqrt{30} \cdot \sqrt{11} }.

3.

\cos\angle 3=\dfrac{\vec{b}\cdot (\vec{a}-\vec{b})}{|\vec{b}|\cdot |\vec{a}-\vec{b}|}=\dfrac{1\cdot 1+2\cdot (-3)+4\cdot 1}{\sqrt{1^2+2^2+4^2}\cdot \sqrt{1^2+(-3)^2+1^2} }=\dfrac{-1}{\sqrt{21} \cdot \sqrt{11} }.

Then

  • \angle 1=\arccos \left(\dfrac{20}{\sqrt{30} \cdot \sqrt{21} }\right)\approx 37.17^{\circ};
  • \angle 2=\arccos \left(\dfrac{10}{\sqrt{30} \cdot \sqrt{11} }\right)\approx 56.60^{\circ};
  • \angle 3=\arccos \left(\dfrac{-1}{\sqrt{21} \cdot \sqrt{11} }\right)\approx 93.77^{\circ}.
You might be interested in
-2x+y=-12, 3y- 2x=-12
inessss [21]

Answer:

x = 6

y = 0

Explanation:

−2x+y=−12;3y−2x=−12

Rewrite equations:

−2x+y=−12;−2x+3y=−12

Solve −2x+y=−12 for y

−2x+y=−12

−2x+y+2x=−12+2x(Add 2x to both sides)

y=2x−12

Substitute 2x−12 for y in −2x+3y=−12

−2x+3y=−12

−2x+3(2x−12)=−12

4x−36=−12 (Simplify both sides of the equation)

4x−36+36=−12+36 (Add 36 to both sides)

4x=24

Divide both sides by 4

4x / 4 = 24 / 4

x = 6

Substitute 6 for x in y=2x−12

y=2x−12

y=(2)(6)−12

y=0 (Simplify both sides of the equation)

6 0
3 years ago
Read 2 more answers
Two number cubes are rolled. What is the probability that the sum of the numbers rolled is either 3 or 9? as a fraction
matrenka [14]
If you are reffering to dice itd be 4/12 simplified is 1/3

6 0
3 years ago
Help on math homework
Tasya [4]
I believe these are the answers but I’m not sure about letter A

3 0
3 years ago
What is the X-coordinate of the Point (6,8)?
klio [65]
6 is the x coordinte
5 0
3 years ago
Read 2 more answers
Please answer this!!! 40 Points, im really desperate. Also, more than one answer here
erastovalidia [21]

Answer:

x = -1, 0

Step-by-step explanation:

The solution to f(x) = g(x) are the x values for which outputs (function values) are equal.

Looking at the data, f(x) and g(x) have same outputs at x = -1 and x = 0

7 0
2 years ago
Other questions:
  • An indirect proof of the statement "If x=2, then √x is not a rational number" coul begin with the assumption that
    7·1 answer
  • How does the value of a decimal change as you move left to right
    10·1 answer
  • If an adult female weighs 50 kg, how many mg of
    5·1 answer
  • MOre geometry.<br> Sorry i struggle.
    8·1 answer
  • Whats the area of a rectangle 17ft 4ft​
    6·2 answers
  • I NEED THIS ASAP!!! What is GC?
    9·1 answer
  • Zero product property:<br><br> (x-4)(-5x+1)=0<br><br> lesser x=<br><br> greater x=
    6·1 answer
  • Express in the form n : 1. Give n as a fully simplified fraction. 10 : 12
    13·1 answer
  • A triangular prism is shown. a cylinder is cut out of the center of the prism. the triangular base has side lengths of 14 and 10
    7·2 answers
  • For a contract that would last for the next 2 years, we are going to be paid at the end of every 4 months where the payments are
    12·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!