Question

If e1 and e2 are two unit vectors and theta is the angle between them then costheta/2

Answer 1

e1 and e2 are two unit vectors and θ is the angle between them.

taking dot product of e1 and e2.

so, e1.e2 = |e1|.|e2|cosθ

⇒e1.e1 = cosθ

[ |e1| = |e2| = 1 as e1 and e2 are unit vectors ]

we know, cos2x = 2cos²x - 1

so, cosθ = 2cos²(θ/2) - 1 = e1.e2

⇒2cos²(θ/2) = e1.e2 + 1

⇒cos²(θ/2) = (e1.e2 + 1)/2

⇒cos(θ/2) = √{(e1.e2 + 1)/2}

hence, cos(θ/2) = √{(e1.e2 + 1)/2}

Answer 2

Step-by-step explanation:

E1 and e2 are two unit vector

angle between them = ∅

we know,

| A - B | = √(| A|² +| B|² - 2|A||B|.cos∅ )

use this ,

|e1 - e2 | =√ ( |e1|² +| e2|² - 2|e1|.|e2|cos∅)

here , e1 and e2 unit vectors

so, |e1 | = 1

|e2 | = 1

|e1 - e2 | = √( 1² + 1² -2.1.1 cos∅)

= √( 2 - 2cos∅)

= √2(1 - cos∅)

= √2×2sin²∅/2

=2sin∅/2

hence, | e1 -e2 | = 2sin∅/2

sin∅/2 = 1/2 | e1 - e2 |

hence, proved ///

[ note :- ( 1 - cos∅) = 2sin²∅/2]

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