Chapter 9: Q. 37 (page 613)

Suppose that vand ware unit vectors. If the angle between vand iis $α$ and that between wand iis $β,$ use the idea of the dot product $v \cdot w$ to prove that
$\cos (α - β) = \cos α \cos β + \sin α \sin β$

Short Answer

Expert verified

It is proved that $\cos (α - β) = \cos α \cos β + \sin α \sin β$ .

Step by step solution

Step 1. Given Information

there are two unit vectors vand w.The angle between vand iis $α$ and that between wand iis $β .$ We have to use the idea of dot product and prove that
$\cos (α - β) = \cos α \cos β + \sin α \sin β .$

Step 2. Proving

If vand ware unit vectors.

So, $||v|| = 1, ||w|| = 1 .$

Now, the dot product of

$v \cdot w = ||v|| ||w|| cosθ v \cdot w = cosθ [||v|| = 1, ||w|| = 1]$

When $θ = α - β o r β - α$ we get,

$v \cdot w = cosθ v \cdot w = \cos (α - β) . . . . . . . . (a)$

Now,

$v = ||v|| (cosα i + sinα j) v = (cosα i + sinα j) [||v|| = 1]$

And

w = ||w|| (\cos β i + \sin β j) w = (\cos β i + \sin β j) [||w|| = 1]

So, $\cos (α - β) = v \cdot w \cos (α - β) = (cosα i + sinα j) \cdot (cosβ i + sinβ j) \cos (α - β) = cosα cosβ + sinα sinβ$

Hence proved.

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Suppose that vand ware unit vectors. If the angle between vand iis $α$ and that between wand iis $β,$ use the idea of the dot product $v \cdot w$ to prove that
$\cos (α - β) = \cos α \cos β + \sin α \sin β$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving

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Suppose that vand ware unit vectors. If the angle between vand iis α and that between wand iis β, use the idea of the dot product v·wto prove thatcosα-β=cosαcosβ+sinαsinβ

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

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Calculus

Study anywhere. Anytime. Across all devices.

Suppose that vand ware unit vectors. If the angle between vand iis $α$ and that between wand iis $β,$ use the idea of the dot product $v \cdot w$ to prove that
$\cos (α - β) = \cos α \cos β + \sin α \sin β$