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Sum and subtract of vectors
Given the vectors $\vec{u}=(3,-2)$ and $\vec{v}=(-1,5)$, determine:
- $3\vec{u}-2\vec{v}$
- $-\vec{u}-\vec{v}$
- $5\vec{u}+2\vec{v}$
- $\vec{u}+3\vec{v}$
Is there one which is a unit vector?
- $3\vec{u}-2\vec{v}=3(3,-2)-2(-1,5)=(9,-6)+(2,-10)=(11,-16)$
- $-\vec{u}-\vec{v}=-(3,-2)-(-1,5)=(-2,-3)$
- $5\vec{u}+2\vec{v}=5(3,-2)+2(-1,5)=(15,-10)+(-2,10)=(13,0)$
- $\vec{u}+3\vec{v}=(3,-2)+3(-1,5)=(0,13)$
$\begin{array}{l} |(11,-16)|=\sqrt{11^2+(-16)^2}=\sqrt{121+256}=\sqrt{377} \ |(-2,-3)|=\sqrt{(-2)^2+(-3)^2}=\sqrt{4+9}=\sqrt{13} \ |(13,0)|=\sqrt{13^2+0^2}=\sqrt{169}=13 \ |(0,13)|=\sqrt{0^2+13^2}=\sqrt{169}=13 \end{array} $
We can see, then, that none of these norms is one. Therefore, none of these vectors are unit vectors.
- $(11,-16)$
- $(-2,-3)$
- $(13,0)$
- $(0,13)$
None of these vectors is a unit vector.