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Sagot :
Answer:
The acute angle lies between the vectors a=3i+4j and b=7i+j is 45°
Explanation:The given vectors are:
a = 3i + 4j
b = 7i + j
The acute angle between vectors a and and b is given by the formula:
[tex]\theta=\cos ^{-1}\frac{a.b}{|a\mleft\Vert b\mright|}[/tex]The scalar product of vectors a and b is:
a.b = (3i + 4j).(7i + j)
a.b = (3x7) + (4x1)
a.b = 21 + 4
a.b = 25
The magnitude of a is:
[tex]\begin{gathered} |a|=\sqrt[]{3^2+4^2} \\ |a|=\sqrt[]{9+16} \\ |a|=\sqrt[]{25} \\ |a|=5 \end{gathered}[/tex]The magnitude of b is:
[tex]\begin{gathered} |b|=\sqrt[]{7^2+1^2} \\ |b|=\sqrt[]{49+1} \\ |b|=\sqrt[]{50} \\ |b|=5\sqrt[]{2} \end{gathered}[/tex]Substituting the values of a.b, |a|, and |b| into the formula for the acute angle.
[tex]\begin{gathered} \theta=\cos ^{-1}\frac{a.b}{|a\Vert b|} \\ \theta=\cos ^{-1}\frac{25}{5\times5\sqrt[]{2}} \\ \theta=\cos ^{-1}\frac{25}{25\sqrt[]{2}} \\ \theta=\cos ^{-1}\frac{1}{\sqrt[]{2}} \\ \theta=45^0 \end{gathered}[/tex]Therefore, the acute angle lies between the vectors a=3i+4j and b=7i+j is 45°
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