Home
Class 11
PHYSICS
Unit vector hat(P) and hat(Q) are inclin...

Unit vector `hat(P)` and `hat(Q)` are inclined at an angle `theta`. Prove that `|hat(P)-hat(Q)|= 2 sin(theta//2)`.

Text Solution

Verified by Experts

`|hat(P)|=|hat(Q)|=1`, because they are unit vectors.
LHS=`|hat(P)-hat(Q)|`
`sqrt((hat(P)-hat(Q)).(hat(P)-hat(Q)))=sqrt((P^(2)+Q^(2)-2hat(P).hat(Q)))`
`=sqrt(1^(2)+1^(2)-2PQ cos theta)=sqrt(2-2xx1xx1 cos theta)`
`sqrt(2(1-cos theta))`
`sqrt(2[2sin^(2)(theta//2)])=2 sin (theta//2)= RHS`
Promotional Banner

Topper's Solved these Questions

  • VECTORS

    CENGAGE PHYSICS|Exercise Exercise Single Correct|51 Videos

  • VECTORS

    CENGAGE PHYSICS|Exercise Exercise Multiple Correct|5 Videos

  • VECTORS

    CENGAGE PHYSICS|Exercise Exercise 3.2|13 Videos

  • TRAVELLING WAVES

    CENGAGE PHYSICS|Exercise Integer|9 Videos

  • WORK, POWER & ENERGY

    CENGAGE PHYSICS|Exercise Archives (integer)|4 Videos

Similar Questions

Explore conceptually related problems

If unit vectors hat(a) and hat(b) are inclined at angle theta , then prove that |vec(a) - vec(b)| = 2 sin.(theta)/(2) .

Unit vectors widehat a and hat b are inclined at an angle 2 theta and |widehat a-hat b|<11 l 0<=theta<=pi?, then theta may belong to

If widehat a and hat b are unit vectors inclined at an angle theta, then prove that cos(theta)/(2)=(1)/(2)|widehat a+hat b|tan(theta)/(2)=(1)/(2)|(widehat a-hat b)/(widehat a+hat b)|

If widehat a and hat b are two unit vectors inclined at an angle theta, then sin((theta)/(2))

If widehat aandhat b are unit vectors inclined at an angle theta, then prove that (sin theta)/(2)=(1)/(2)|widehat a-hat b|

If widehat a and hat b are unit vectors inclined at an angle theta then prove that (cos theta)/(2)=(1)/(2)|widehat a+hat b|(tan theta)/(2)=(|widehat a-hat b|)/(|widehat a+hat b|)

If hat(i), hat(j) and hat(k) are unit vectors along x,y and z-axis respectively, the angle theta between the vector hat(i) + hat(j) + hat(k)" ""and vector" hat(j) is given by

Let hat(a), hat(b) be two unit vectors and 0 be the angle between them. What is sin ((theta)/(2)) equal to

If hat(i), hat(j) and hat(k) represent unit vectors along the x, y and z axes respectively, then the angle theta between the vectors (hat(i) + hat(j) + hat(k)) and (hat(i) + hat(j)) is equal to :

CENGAGE PHYSICS-VECTORS-Exercise Subjective
  1. Can you find at least one vector perpendiculr to 3hat(i)-4hat(j)+7hat(...

    Text Solution

    |

  2. Two forces P and Q acting at a point are such that if P is reveserd, t...

    Text Solution

    |

  3. Unit vector hat(P) and hat(Q) are inclined at an angle theta. Prove th...

    Text Solution

    |

  4. A sail boat sails 2km due east, 5km 37^(@) south of east, and finally ...

    Text Solution

    |

  5. Two forces of magnitudes P and Q are inclined at an angle (theta). The...

    Text Solution

    |

  6. A vector vec(B) which has a magnitude 8.0 is added to a vector vec(A) ...

    Text Solution

    |

  7. Three vector as shown in (figure) have magnitudes |vec(a)|=3,|vec(b)|=...

    Text Solution

    |

  8. A bouy is attached to three tugboats by three ropes. The tugboats are ...

    Text Solution

    |

  9. Two horizontal forces of magnitudes of 10N and P N act on a particle. ...

    Text Solution

    |

  10. The position vectors of two balls are given by vec(r )(1)=2 (m)i+7(m...

    Text Solution

    |

  11. A particle whose speed is 50ms^(-1) moves along the line from A(2,1) t...

    Text Solution

    |

  12. A particle travels with speed 50ms^(-1) from the point (3,-7) in a dir...

    Text Solution

    |

  13. A particle has an initial velocity of 3hat(i) + 4 hat(j) and an accele...

    Text Solution

    |

  14. Forces X,Y and Z have magnitudes 10N 5(sqrt(3)-1) N and 5(sqrt(3)+1) N...

    Text Solution

    |

  15. A particle of m=5kg is momentarily at rest at x= at t=. It is acted up...

    Text Solution

    |

  16. A spy plane is being tacked by a radar. At t=0, its position is report...

    Text Solution

    |

  17. a. Calculate vec(r )=vec(a)-vec(b)+vec(c ), where vec(a)=5hat(i)+4hat(...

    Text Solution

    |

  18. A force vec(F)=3hat(i)+2hat(j)+chat(k)N causes a displacement vec(r )=...

    Text Solution

    |

  19. When two vectors of magnitudes P and Q are inclined at an angle theta,...

    Text Solution

    |

  20. Find the magnitudes of the unknown forces if the sum of all forces is ...

    Text Solution

    |