Is it possible to solve this problem without x-axis and y-axis?

Question

Is it possible to solve this problem without x-axis and y-axis ? And If it is, What is the way to solve it?

Answer 1

Yes, it is possible to solve this problem without defining coordinates. Here is a link that will be of use to you: adding and subtracting vectors geometrically

Answer 2

We don't need axis, the grid is sufficient, as we may add vectors geometrically - by moving them into positions, where the end point of the first vector coincides with the start point of the second one:

Substacting the vector $\overrightarrow C$ is the same as adding the vector $-\overrightarrow C.$

The grid shows us that $\left\lVert \overrightarrow A + \overrightarrow B - \overrightarrow C \right\rVert = 5$ , i. e. the $(c)$ variant of your task is the correct one.

Answer 3

Coordinates of a vector don't depend from the absolute positions of its start and end point, but only by the position of its end point relative to its start point.

Formally,

$$\overrightarrow {XY} = Y - X$$

i. e. if $X=(x_1, x_2)\ $ and $\ Y = (y_1, y_2)$, then

$$\overrightarrow {XY} = (y_1-x_1, \;y_2-x_2)$$

Informally, we obtain coordinates of the vector as horizontal and vertical distance from X to Y:

So we don't need the axes, because from you picture we see that

\begin{align} \overrightarrow {A} &= (3, 0)\\ \overrightarrow {B} &= (1, -3)\\ \overrightarrow {C} &= (4, 2) \end{align}

so

$$\left\lVert \overrightarrow A + \overrightarrow B - \overrightarrow C \right\rVert = \left\lVert (3, 0) + (1, -3) - (4, 2) \right\rVert = \left\lVert (0, -5) \right\rVert = 5$$