2. Which of the following are true and which are false?(I) For every u and the vector projection of u onto is equal to the vector projectionof u onto 20.11(II) (i+j) × (-3)=0 where = (1,0,0) and 7 = (0,1,0) -(III) (xv) = 0 for every pair of vectors i and 7..140

(I). Given that u→ & v→ are two vectors let, u→= xi^+yj ^ and v→= ai^+bj^ therefore,…

Answered: 2. Which of the following are true and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2. Which of the following are true and which are false?
(I) For every u and the vector projection of u onto is equal to the vector projection
of u onto 20.
11
(II) (i+j) × (-3)=0 where = (1,0,0) and 7 = (0,1,0) -
(III) (xv) = 0 for every pair of vectors i and 7.
.
140

Expert Solution

Step 1

(I).

$G i v e n t h a t \vec{u} & \vec{v} a r e t w o v e c t o r s$

$l e t, \vec{u} = x \hat{i} + y \hat{j} a n d \vec{v} = a \hat{i} + b \hat{j}$

$t h e r e f o r e, t h e p r o j e c t i o n o f \vec{u} o n t o \vec{v} i s -$

$p r o j_{v} \vec{u} = (\frac{\vec{u} . \vec{v}}{{||\vec{v}||}^{2}}) \vec{v}$

= $(\frac{a x + b y}{a^{2} + b^{2}}) . (a \hat{i} + b \hat{j)}$

$n o w, i f w e t a k e \vec{u} a n d 2 \vec{v},$

$t h e n u = x \hat{i} + y \hat{j} a n d 2 \vec{v} = 2 a \hat{i} + 2 b \hat{j}$

$t h e r e f o r e, t h e p r o j e c t i o n o f \vec{u} o n t o 2 \vec{v} i s -$

$p r o j_{2 v} \vec{u} = (\frac{\vec{u} . \vec{2 v}}{{||\vec{2 v}||}^{2}}) \vec{2 v}$

$= (\frac{2 a x + 2 b y}{4 a^{2} + 4 b^{2}}) \cdot (2 a \hat{i} + 2 b \hat{j)}$

$= (\frac{a x + b y}{a^{2} + b^{2}}) \cdot (a \hat{i} + b \hat{j)} = p r o j_{v} \vec{u}$

Therefore, for every $\vec{u}$ and $\vec{v}$ the vector projection of $\vec{u}$ onto $\vec{v}$ is equal to the vector projection of $\vec{u}$ onto $\vec{2 v}$ .

so, given statement is TRUE.

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2. Which of the following are true and which are false? (I) For every u and the vector projection of u onto is equal to the vector projection of u onto 20. (II) (i + j) × (i − 3) =0 where = (1,0,0) and 7 = (0,1,0) - (III) (xv) = 0 for every pair of vectors u and 7. 140