Vectors - Orthogonal

Bit stuck on exercise 10. Attached my working so far, but idk how to find all of the orthogonal vectors. Could anyone help please?

So how is a related to b (or vice versa) and what is the general form of the set of orthogonal vectors ...
It helps to have an idea of what sort of answer youre expecting in order to know what to do with the algebra. In this case its going to be a line so the direction is important, but the magnitude not.

I don’t really know how to answer these questions, we haven’t covered anything about sets of vectors and I’m so confused. Is it something to do with a and b having the same direction? And I’m assuming in an orthogonal set all of the vectors are orthogonal to each other?

a and b are the coefficients of the vector and the direction of that vector is atan(b/a) so its the ratio of a and b thats important. So from your equation you can write
b = ... a
and that gives the vectors as you have the direction which is perpendicular to the given vector. Tbh, you could pretty much write it down using a simple gcse maths result which says the product of perpendicular vectors gradients is -1.

As before, you need to understand its the direction thats important here, and varying the magnitude gives the family of all orthogonal vectors. Note that as its effectively 2D, you could sketch the vector and guestimate all the orthogonal ones to get the insight.

Okay so b = -3a so it’s a 1:-3 ratio. For my original vector the gradient in xy plane would be 9/3 = 3 and for my new vector the gradient would be -3/1 = -3 so they’re perpendicular. Is that right?

How would I write the final answer? Would it be all vectors (a, -3a, 0) for all values a or is that wrong?

Should be fine.

Can also say the span of (1,-3,0) is the set of all vectors orthogonal to v.

If the question asked for the orthonormal vector, then a would need to be a precise value.