Question

if unit vectors A and B cabs are inclined at angle theta then prove that a cap
minus b cap is equal to 2 sin theta upon 2

Answer 1

Since a and b are unit vectors |a| = |b| =1

Let angle be x between vectors a and b (just fr convenience while typing ;-) )

|a-b| = {|a|^2 +|b|^2 + 2 |a||b|cos(180- x)}^0.5

Putting values in rhs

|a-b| = {1+1+2*1*1*(-cos x)}^0.5

= {2–2cosx}^0.5

={2(1-cosx)}^0.5

={2*2(sinx/2)^2}^0.5

= 2 sinx/2

|a-b|= 2 sinx/2

1/2|a-b|= sinx/2

Hence, proved.

Answer 2

Answer:

Given A and B are unit vectors.

$|A| = | B| = 1$

Let angle be p between vectors A and B

$∣A − B∣ =\sqrt{ ∣A∣² + ∣B∣² + 2∣A∣∣B∣cos(180 − x)}$

Putting values in above equation,

$|A-B| = \sqrt{1 + 1 + 2 \times 1 \times 1 \times 1 \cos(p)}$

$= \sqrt{2 - 2 \cos(p) }$

$= \sqrt{2(1 - \cos(p) )}$

$= 2 \sin( \frac{p}{2} )$

$|A-B| = 2 \sin( \frac{p}{2} )$

$\frac{1}{2} |A-B| = \sin( \frac{p}{2} )$

Left hand equal to right hand (Proved)