SOLUTION: u =−6i − 2j v= −9i + 4j Find the degree measure of the angle 𝜃 between the vectors. (Round your answer to two decimal places.) 𝜃 =_____°

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Question 1173623: u =−6i − 2j
v= −9i + 4j
Find the degree measure of the angle 𝜃 between the vectors. (Round your answer to two decimal places.)
𝜃 =_____°
Found 3 solutions by mccravyedwin, Edwin McCravy, ikleyn:
Answer by mccravyedwin(407) About Me  (Show Source):
You can put this solution on YOUR website!
Plug in the formula for the angle between vectors u and v where:

u = x1i + y1j
v = x2i + y2j

theta%22%22=%22%22%22tan%22%5E%28-1%29abs%28%28x%5B1%5Dy%5B2%5D-x%5B2%5Dy%5B1%5D%29%2F%28x%5B1%5Dx%5B2%5D%2By%5B1%5Dy%5B2%5D%29%29

u = -6i - 2j
v = -9i + 4j

theta%22%22=%22%22%22tan%22%5E%28-1%29abs%28%28%28-6%29%284%29-%28-9%29%28-2%29%29%2F%28%28-6%29%28-9%29%2B%28-2%29%284%29%29%29

theta%22%22=%22%22%22tan%22%5E%28-1%29abs%28%28%28-24%29-%2818%29%29%2F%28%2854%29%2B%28-8%29%29%29

theta%22%22=%22%22%22tan%22%5E%28-1%29abs%28%28-42%29%2F%2846%29%29

theta%22%22=%22%22%22tan%22%5E%28-1%29abs%28-0.9130434783%29

theta%22%22=%22%22%22tan%22%5E%28-1%29%280.9130434783%29

theta%22%22=%22%2242.3974378%5Eo

theta%22%22=%22%22%2242.40%22%5Eo

Edwin

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
You can also do it this way:

cos%28theta%29%22%22=%22%22%28u%2Av%29%2F%28%22%7Cu%7C%7Cv%7C%22%29

cos%28theta%29%22%22=%22%22

cos%28theta%29%22%22=%22%22%2854%2B%28-8%29%29%2F%28sqrt%2836%2B4%29sqrt%2881%2B16%29%29%29

cos%28theta%29%22%22=%22%22%2846%29%2F%28sqrt%2840%29sqrt%2897%29%29

cos%28theta%29%22%22=%22%22%2846%29%2Fsqrt%2838800%29

cos%28theta%29%22%22=%22%220.7384854939

theta%22%22=%22%2242.3974378%5Eo

theta%22%22=%22%22%2242.40%22%5Eo

Edwin

Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.

You just got two solutions from Edwin,  and you,  probably,  have a question,  which approach/method to prefer.


Of these two approaches,  the method using cosine is  ROBUST  and works  ALWAYS,  and  ALWAYS  produces correct results.


The Edwin's formulas with tan and arctan have a huge underwater stone.


In this form, as they presented,  they work and they produce a correct answer for an  ACUTE  angle,  ONLY.


For an  OBTUSE  angle they do not work.


AGAIN:   the method using cosine and scalar products,  is  ROBUST  and  STANDARD  method for such problems.


For them,  use it and  ONLY  it . . .

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    - Introduction to dot-product
    - Formula for Dot-product of vectors in a plane via the vectors components
    - Dot-product of vectors in a coordinate plane and the angle between two vectors (*)
    - Solved problems on Dot-product of vectors and the angle between two vectors

    - HOW TO find dot-product of two vectors in a plane
    - HOW TO find scalar product of two vectors in a coordinate plane
    - HOW TO find the angle between two vectors in a coordinate plane


Of these lessons, the key is the lesson marked (*) in the list.
It is the most relevant lesson to your problem.

Other lessons illustrate how everything work.


Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic
"Dot-product for vectors in a coordinate plane".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.