pedzenekO

2021-08-19

Two vectors, a and b, are unit vectors with an angle of 60 degrees between them (when tail-to-tail).
If the vectors below are orthogonal, what is the value(s) of m?
u= a-3b
v=ma + b

Given:
The vectors $u=a-3b$ and $v=ma+b$ are orthogonal.
The vectors a and b are unit vectors with an angle of 60 degrees between them.
As u and v are orthogonal vectors, therefore, $u·v=0$.
$\left(a-3b\right)\left(ma+b\right)=a\left(ma+b\right)-3b\left(ma+b\right)=$
=$m{a}^{2}+a\cdot b-3mb\cdot a-3{b}^{2}$
As a and b are unit vectors, therefore, |a|=1, |b|=1
As $a\cdot b=|a|\cdot |b|\mathrm{cos}\theta$
$\left(a-3b\right)\left(ma+b\right)=m\left(1\right)+|a||b|\mathrm{cos}\left(60\right)-3m|b||a|\mathrm{cos}\left(60\right)-3\left(1\right)=m+\left(1\right)\left(1\right)\left(\frac{1}{2}\right)-3m\left(1\right)\left(1\right)\left(\frac{1}{2}\right)-3=m+\frac{1}{2}-\frac{3m}{2}-3$
Therefore,
$u\cdot v=-\frac{m}{2}-\frac{5}{2}=0$
Simplifying the above equation,
$-\frac{m}{2}-\frac{5}{2}=0$
$-\frac{m}{2}=\frac{5}{2}$
Multiply both sides by 2,
-m=5
m=-5

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