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To the nearest tenth of a degree, find the size of the angle formed by placing the vectors [4, 0] and [−6, 5] tail-to-tail at the origin.
It is understood in questions such as this that the answer is smaller than 180 degrees.
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If "a" and "b" are two vectors in a plane and is the angle between them, then
= ,
where (a*b) is the scalar product of the vectors "a" and "b" and |a| and |b| are their lengths.
This formula is VERY WELL known in Math.
You can find all the details, proof and examples of using this formula in the lessons referred at the end of my post.
In the given problem, a = [4,0], b = [-6,5];
|a| = = = 4; |b| = = = ;
(a,b) = 4*(-6) + 0*5 = - 24.
Therefore, = = - = -0.768221,
= arccos(-0.768221) = use your calculator = 2.446854 radians = 140.9545 degrees ~= 141.0 degrees (rounded as requested). ANSWER
Solved, explained and completed.
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As promised, you can find all the details, proof and examples of using this formula in my lessons
- 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
- Perpendicular vectors in a coordinate plane
- Solved problems on Dot-product of vectors and the angle between two vectors
- Properties of Dot-product of vectors in a coordinate plane
- The formula for the angle between two vectors and the formula for cosines of the difference of two angles
in this site.
There are closely related short lessons of the "HOW TO . . . " type on Dot-product:
- 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
For the full list of my lessons on dot-product with short annotations see the file OVERVIEW of lessons on Dot-product.
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.
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