kariboucnp

2022-12-31

can the vector components be negative

abbrechtq17

Expert

Step 1
Yes, a vector component may be negative. It depends on along which direction we wish to calculate that component.
If a vector $A=2i+3j$, find its components along $i+j$ and $i-j$
Component of $\stackrel{\to }{A}$ along $\stackrel{\to }{B}$ is $\stackrel{\to }{A}\cdot \stackrel{^}{B}$ where $\stackrel{^}{B}=\frac{\stackrel{\to }{B}}{|\stackrel{\to }{B}|}$ is a unit vector along B
Step 2
Component Along $\stackrel{^}{i}+\stackrel{^}{j}$ is $\stackrel{\to }{A}\cdot \frac{\left(\stackrel{^}{i}+\stackrel{^}{j}\right)}{\sqrt{{1}^{2}+{1}^{2}}}=\left(2\stackrel{^}{i}+3\stackrel{^}{j}\right)\cdot \frac{\left(\stackrel{^}{i}+\stackrel{^}{j}\right)}{\sqrt{2}}$
$=\frac{2+3}{\sqrt{2}}=\frac{5}{\sqrt{2}}$
Component along $\stackrel{^}{i}-\stackrel{^}{j}$ is $A\cdot \frac{\left(\stackrel{^}{i}-\stackrel{^}{j}\right)}{\sqrt{2}}=\left(2\stackrel{^}{i}+3\stackrel{^}{j}\right)\cdot \frac{\left(\stackrel{^}{i}-\stackrel{^}{j}\right)}{\sqrt{2}}$
$=\frac{2-3}{\sqrt{2}}=-\frac{1}{\sqrt{2}}$

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