Operations with Algebraic Vectors in R2

Question

I was given a thinking question as my homework.

Here is the question

So basically what I've tried to do is: Since it's given that both the vectors start from the exact same point, and have the same magnitude, I tried to find the magnitude of PB by calculating the distance between vector P and B (a,0) + (0,2) and got sqrt((-a)^2 + 2

Following the same procedure for vector PA, I got sqrt((5-a)^2 - 0)

Now, I tried making both magnitudes equal to each other and solve for a since I know both vectors are to have the same magnitude, however, my result is different than the one in the textbook.

So my question is: am I missing anything? Does my logic even work here?

Answer 1

The distance between two points $(a_1, a_2)$ and $(b_1, b_2) \in \mathbb{R}^2$ is given by:

$$\sqrt{(b_1 - a_1)^2 + (b_2 - a_2)^2}$$

For the first part, we have

$$|\overrightarrow {PA}| = |\overrightarrow {PB}|$$

$$\sqrt{a^2 + 2^2} = \sqrt{(5-a)^2}$$

$$a^2 + 4 = 25 -10a + a^2$$

$$a = 2.1$$

So $P$ is the point $(2.1,0)$.

For the second part, we have

$$|\overrightarrow{QA}| = |\overrightarrow{QB}|$$

$$\sqrt{(b-2)^2} = \sqrt{b^2 + 5^2}$$

$$b^2 -4b + 4 = b^2 + 25$$

$$b = \frac{-21}{4} = -5.25$$

So $Q$ is the point $(0,-5.25)$.