Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
University Physics Volume 1

Challenge Problems

University Physics Volume 1Challenge Problems
Search for key terms or text.

Challenge Problems

88.

Vector B→B→ is 5.0 cm long and vector A→A→ is 4.0 cm long. Find the angle between these two vectors when |A→+B→|=3.0cm|A→+B→|=3.0cm.

89.

What is the component of the force vector G→=(3.0i^+4.0j^+10.0k^)NG→=(3.0i^+4.0j^+10.0k^)N along the force vector H→=(1.0i^+4.0j^)NH→=(1.0i^+4.0j^)N?

90.

The following figure shows a triangle formed by the three vectors A→A→, B→B→, and C→C→. If vector C→′C→′ is drawn between the midpoints of vectors A→A→ and B→B→, show that C→′=C→/2C→′=C→/2.

91.

Distances between points in a plane do not change when a coordinate system is rotated. In other words, the magnitude of a vector is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle φφ to become a new coordinate system S′S′, as shown in the following figure. A point in a plane has coordinates (x, y) in S and coordinates (x′,y′)(x′,y′) in S′S′.

(a) Show that, during the transformation of rotation, the coordinates in S′S′ are expressed in terms of the coordinates in S by the following relations:

{ x ′ = x cos φ + y sin φ y ′ = − x sin φ + y cos φ . { x ′ = x cos φ + y sin φ y ′ = − x sin φ + y cos φ .

(b) Show that the distance of point P to the origin is invariant under rotations of the coordinate system. Here, you have to show that

x 2 + y 2 = x ′ 2 + y ′ 2 . x 2 + y 2 = x ′ 2 + y ′ 2 .

(c) Show that the distance between points P and Q is invariant under rotations of the coordinate system. Here, you have to show that

( x P − x Q ) 2 + ( y P − y Q ) 2 = ( x P − x Q ) 2 + ( y P − y Q ) 2 =
( x ' P − x ' Q ) 2 + ( y ' P − y ' Q ) 2 . ( x ' P − x ' Q ) 2 + ( y ' P − y ' Q ) 2 .
PreviousNext
Order a print copy
Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:

    Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction

  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:

    Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction

Citation information

© Jul 8, 2025 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.