There are infinitely many vectors orthogonal to 24i+7j as if v is some vector orthogonal to it, then kv is also orthogonal for any scalar k. So I'm guessing you're missing some constraint on the length of the vector like it must be the same length as 24i+7j or maybe that it should be length 1. So what I'm going to do is find one vector orthogonal to it and then you can scale it as necessary to meet the extra condition.

By definition, vectors uu and vv are orthogonal if and only if

u⋅v=0

So let's say v=ai+bj is orthogonal to 24i+7j. Then we see that

\(\displaystyle{\left({24}{i}+{7}{j}\right)}⋅{v}={0}\)

\(\displaystyle{\left({24}{i}+{7}{j}\right)}⋅{\left({a}{i}+{b}{j}\right)}={0}\)

24a+7b=0

At this point, we can find any solution. Let's take a=−7. Substituting that in we then find that

24(−7)+7b=0

−168+7b=0

7b=168

b=24

Hence v=−7i+24j is one vector that's orthogonal to 24i+7j, as promised.

By definition, vectors uu and vv are orthogonal if and only if

u⋅v=0

So let's say v=ai+bj is orthogonal to 24i+7j. Then we see that

\(\displaystyle{\left({24}{i}+{7}{j}\right)}⋅{v}={0}\)

\(\displaystyle{\left({24}{i}+{7}{j}\right)}⋅{\left({a}{i}+{b}{j}\right)}={0}\)

24a+7b=0

At this point, we can find any solution. Let's take a=−7. Substituting that in we then find that

24(−7)+7b=0

−168+7b=0

7b=168

b=24

Hence v=−7i+24j is one vector that's orthogonal to 24i+7j, as promised.