Question

Let 𝑎⃗ and 𝑏 be two vectors in the xy plane making angles 𝜃 and ∅ with the x axis, respectively.

Use vector algebra to show that:
cos(𝜃 − ∅) = cos 𝜃 cos ∅ + sin 𝜃 sin ∅

Answer 1

good

Answer 2

[AI-generated answer — Explain Bot]

To use vector algebra to show that cos(𝜃 − ∅) = cos 𝜃 cos ∅ + sin 𝜃 sin ∅, we can express the vectors 𝑎⃗ and 𝑏 in terms of their components.

Let's assume that 𝑎⃗ = 𝑎𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 and 𝑏 = 𝑏𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒, where 𝑎 and 𝑏 represent the magnitudes of the vectors 𝑎⃗ and 𝑏, respectively.

To find the components of 𝑎⃗ , we can use the cosine and sine functions. Since 𝑎⃗ makes an angle 𝜃 with the x-axis, its x-component can be found using the equation:
𝑎_𝑥 = 𝑎 cos 𝜃

Similarly, the y-component of 𝑎⃗ can be found using the equation:
𝑎_𝑦 = 𝑎 sin 𝜃

Using the same logic, the components of 𝑏 can be found as:
𝑏_𝑥 = 𝑏 cos ∅
𝑏_𝑦 = 𝑏 sin ∅

Now, let's consider the dot product of 𝑎⃗ and 𝑏, denoted as 𝑎⃗ · 𝑏.

𝑎⃗ · 𝑏 = (𝑎_𝑥 𝑏_𝑥) + (𝑎_𝑦 𝑏_𝑦)

Substituting the components of 𝑎⃗ and 𝑏, we get:

𝑎⃗ · 𝑏 = (𝑎 cos 𝜃)(𝑏 cos ∅) + (𝑎 sin 𝜃)(𝑏 sin ∅)

Expanding the equation, we get:

𝑎⃗ · 𝑏 = 𝑎𝑏 cos 𝜃 cos ∅ + 𝑎𝑏 sin 𝜃 sin ∅

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:

𝑎⃗ · 𝑏 = 𝑎𝑏 cos(𝜃 - ∅)

Setting it equal to the previous expression, we have:

𝑎𝑏 cos(𝜃 - ∅) = 𝑎𝑏 cos 𝜃 cos ∅ + 𝑎𝑏 sin 𝜃 sin ∅

Now, dividing both sides of the equation by 𝑎𝑏, we get:

cos(𝜃 - ∅) = cos 𝜃 cos ∅ + sin 𝜃 sin ∅

And there you have it! We have used vector algebra to show that cos(𝜃 - ∅) is equal to cos 𝜃 cos ∅ + sin 𝜃 sin ∅.

Answer 3

The angle between a and b is α = θ-Ø

a•b = |a| |b| cosα

a•b = (xa*xb)+(ya*yb)
xa = |a| cosθ
ya = |a| sinθ
...
Carry on, and you will see the result fall right out.