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University Physics with Modern Physics

Hugh D. Young, Roger A. Freedman

Physics

14th edition Edition 路 1596 pages

ISBN: 9780321973610

Chapter 1: Q45 E (page 1)

Find the angle between each of these pairs of vectors:
(a) $\overset{鈫�}{A} = - 2.00 i + 6.00 j$ and $\overset{鈫�}{B} = 2.00 \hat{i}$ $- 3.00 \hat{j}$
(b) $\overset{鈫�}{A} = 3.00 \hat{i} + 5.00 \hat{j}$ and $\overset{鈫�}{B} = 10.00 \hat{i}$ $+ 6.00 \hat{j}$
(c) $\overset{鈫�}{A} = - 4.00 \hat{i}$ $+ 2.00 \hat{j}$ and $\overset{鈫�}{B} = 7.00 \hat{i}$ $+ 14.00 \hat{j}$

Short Answer

Expert verified

The angles are $\cos^{- 1} (\frac{- 22}{\sqrt{40} \sqrt{13}}), \cos^{- 1} (\frac{60}{\sqrt{34} \sqrt{136}})$ and 0.

Step by step solution

01

Given data

$\overset{鈬赌}{A}$ $= - 2.00 \hat{喂}$ $+ 6.00 \hat{J}$ and $\overset{鈬赌}{B}$ $= 2.00 \hat{喂}$ $- 3.00 \hat{J}$

$\overset{鈬赌}{A}$ $= 3.00 \hat{喂}$ $+ 5.00 \hat{J}$ and $\overset{鈫�}{B}$ $= 10.00 \hat{喂}$ + data-custom-editor="chemistry" $6.00 \hat{J}$

$\overset{鈫�}{A}$ $= - 4.00 \hat{喂}$ $+ 2.00 \hat{J}$ and $\overset{鈬赌}{B}$ $= 7.00 \hat{喂}$ $+ 14.00 \hat{J}$

02

(a) The angle between A→and B→

$\overset{鈫�}{A} 路 \overset{鈫�}{B}$ $= (- 2 脳 2) + (6 脳 (- 3)) = - 22$

$A = \sqrt{2^{2} + 6^{2}}$ $= \sqrt{40}$ $= 6.32$

$B = \sqrt{2^{2} + 3^{2}}$ $= \sqrt{13}$ $= 3.6$

role="math" localid="1663683640076" $\cos 胃 = \frac{\overset{鈫�}{A} . \overset{鈫�}{B}}{A B} = \frac{60}{\sqrt{34} \sqrt{136}}$

$胃 = \cos^{- 1} (\frac{60}{\sqrt{34} \sqrt{136}})$

03

(b) The angle between A→ and B→

$\overset{鈫�}{A} . \overset{鈫�}{B} = (- 4 脳 7) + (2 脳 14) = 0$

$A = \sqrt{4^{2} + 2^{2}} = \sqrt{20}$

$B = \sqrt{7^{2} + 14^{2}} = \sqrt{245}$

$\cos 胃 = \frac{\overset{鈫�}{A} . \overset{鈫�}{B}}{A B} = \frac{- 22}{\sqrt{40} \sqrt{13}}$

$胃 = \cos^{- 1} (\frac{- 22}{\sqrt{40} \sqrt{13}})$

04

(a) The angle between A→and B→

$\overset{鈫�}{A} . \overset{鈫�}{B}$ $= (- 4 脳 7) + (2 脳 14) = 0$

$A = {\sqrt{4^{2} + 2^{2}}}^{} = \sqrt{20} B = \sqrt{7^{2} + 14^{2}} = \sqrt{245}$

$\cos 胃 = \frac{\overset{鈫�}{A} . \overset{鈫�}{B}}{A B} = \frac{0}{\sqrt{20} \sqrt{245}} = 0$

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