Problems from Product of a real number by a vector

Given the vectors $$\vec{u}=(2,-2)$$ and $$\vec{v}=(1,3)$$, determine:

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Development:

  • $$3\vec{u}-2\vec{v}=3(2,-2)-2(1,-3)=(6,-6)+(-2,6)=(4,0)$$
  • $$-\vec{u}-\vec{v}=-(2,-2)-(1,-3)=(-3,5)$$
  • $$5\vec{u}+2\vec{v}=5(2,-2)+2(1,-3)=(10,-10)+(2,-6)=(12,-16)$$
  • $$\vec{u}+3\vec{v}=(2,-2)+3(1,-3)=(5,-11)$$

    $$\begin{array}{l} |(4,0)|=\sqrt{4^2+0^2}=\sqrt{16}=\sqrt{4^2}=4 \\ |(-3,5)|=\sqrt{(-3)^2+5^2}=\sqrt{9+25}=\sqrt34 \\ |(12,-16)|=\sqrt{12^2+(-16)^2}=\sqrt{144+256}=\sqrt{400}=\sqrt{20^2}=20 \\ |(5,-11)|=\sqrt{5^2+(-11)^2}=\sqrt{25+121}=\sqrt{146} \end{array} $$

    We can see, then, that none of these norms is one. Therefore, none of these vectors are unit vectors.

Solution:

  • $$(4,0)$$
  • $$(-3,5)$$
  • $$(12,-16)$$
  • $$(5,-11)$$

    None of these vectors is a unit vector.

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