Question

Vectors u and v are orthogonal. If u=<3, 1+b> and v=<5, 1-b> find all possible values for b.

Vectors and spaces
ANSWERED
asked 2021-05-03

Vectors u and v are orthogonal. If \(u=\langle3, 1+b\rangle\) and \(v=\langle5, 1-b\rangle\) find all possible values for b.

Answers (1)

2021-05-04

Remember that two vectors uu and vv are orthogonal if and only if
\(\displaystyle{u}⋅{v}={0}\)
So let's substitute in the expressions from the question and see what we end up with.
\(\displaystyle⟨{3},{1}+{b}⟩⋅⟨{5},{1}−{b}⟩={0}{\left({3}\right)}{\left({5}\right)}+{\left({1}+{b}\right)}{\left({1}−{b}\right)}={0}\)
\((3)(5)+(1+b)(1−b)=0\)
\(\displaystyle{15}+{1}−{b}^{{2}}={0}\)
\(\displaystyle{b}=±{4}\)
So the two possible values of bb are 4 and −4.

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