- Show that
cosθ=(u⋅v)/(|u||v|)for 2D vectors u and v.
(it is quite hard to come up with these challenges, so if you got any ideas, please post them)
this post was submitted on 25 May 2024
32 points (100.0% liked)
Daily Maths Challenges
189 readers
1 users here now
Share your cool maths problems.
Complete a challenge:
- Post your solution in comments, if it is exactly the same as OP's solution, let us know.
- Have fun.
Post a challenge:
- Doesn't have to be original, as long as it is not a duplicate.
- Challenges not riddles, if the post is longer than 3 paragraphs, reconsider yourself.
- Optionally include solution in comments, let it be clear this is not a homework help forums.
- Tag [unsolved] if you don't have a solution yet.
- Please include images, if your question includes complex symbols, attach a render of the maths.
Feel free to contribute to a series by DMing the OP, or start your own challenge series.
founded 6 months ago
MODERATORS
you are viewing a single comment's thread
view the rest of the comments
view the rest of the comments
A good exercise to dust off trig. Hopefully no typos involving u, v, 1, 2, α, and β.
Trigonometric solution
By definition of |u| and Pythagoras, we can see that u1 and u2 can represent the lengths the legs of a right triangle with |u| being the length of the hypotenuse. Similarly for v. The triangles can be visualized in the usual way by plotting the vectors in the XY plane centered at the origin, with u1 and v1 constituting the X coordinates and u2 and v2 the Y coordinates of u and v, respectively.
Let α and β be the respective angles that u and v make with the X-axis. Then, θ = β - α, and
u1 = |u| cos α
u2 = |u| sin α
v1 = |v| cos β
v2 = |v| sin β
Now consider the expression:
|u| |v| cos θ
= |u| |v| cos (β - α)
= |u| |v| (cos β cos α + sin β sin α) (cosine angle difference formula)
= (|u| |v| cos β cos α) + (|u| |v| sin β sin α) (distribute)
= (|u| cos α)(|v| cos β) + (|u| sin α)(|v| sin β) (commutativity, associativity)
= (u1)(v1) + (u2)(v2) (substituting)
= u • v (definition)
visible text
spoilered invisible textmuch obliged