Let πβ and π be two vectors in the xy plane making angles π and β with the x axis, respectively.
Use vector algebra to show that:
cos(π β β
) = cos π cos β
+ sin π sin β
good
To use vector algebra to show that cos(π β β ) = cos π cos β + sin π sin β , we can express the vectors πβ and π in terms of their components.
Let's assume that πβ = πππππππ‘π’ππ and π = πππππππ‘π’ππ, where π and π represent the magnitudes of the vectors πβ and π, respectively.
To find the components of πβ , we can use the cosine and sine functions. Since πβ makes an angle π with the x-axis, its x-component can be found using the equation:
π_π₯ = π cos π
Similarly, the y-component of πβ can be found using the equation:
π_π¦ = π sin π
Using the same logic, the components of π can be found as:
π_π₯ = π cos β
π_π¦ = π sin β
Now, let's consider the dot product of πβ and π, denoted as πβ Β· π.
πβ Β· π = (π_π₯ π_π₯) + (π_π¦ π_π¦)
Substituting the components of πβ and π, we get:
πβ Β· π = (π cos π)(π cos β
) + (π sin π)(π sin β
)
Expanding the equation, we get:
πβ Β· π = ππ cos π cos β
+ ππ sin π sin β
Since the dot product between two vectors is equal to the product of their magnitudes and the cosine of the angle between them, we can rewrite the equation as:
πβ Β· π = ππ cos(π - β
)
Setting it equal to the previous expression, we have:
ππ cos(π - β
) = ππ cos π cos β
+ ππ sin π sin β
Now, dividing both sides of the equation by ππ, we get:
cos(π - β
) = cos π cos β
+ sin π sin β
And there you have it! We have used vector algebra to show that cos(π - β
) is equal to cos π cos β
+ sin π sin β
.
The angle between a and b is Ξ± = ΞΈ-Γ
aβ’b = |a| |b| cosΞ±
aβ’b = (xa*xb)+(ya*yb)
xa = |a| cosΞΈ
ya = |a| sinΞΈ
...
Carry on, and you will see the result fall right out.