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

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

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.