Hello Percy Jackson! 

I don't think its possible to visualize vectors without trigonmetry, but here is my explanation all about vectors below: 

According to my understanding, a "vector" is any quantity, such as force, that has both a magnitude (amount) and a direction. Some common examples of vector quantities are force, displacement, velocity, and acceleration.e.t.c

so... what if you have several vectors and need to add them together? The sum of two vectors is known as the "resultant", and you can use trigonometry to help you find it. 

if your vectors are to form right angle e.g:

As you know that Vectors can be represented graphically using an arrow and so on, the length of the arrow should correspond to the magnitude of the vector, and the direction the arrow is drawn should correspond to the direction of the vector.

To add two vectors graphically, first draw one of the vectors on a piece of paper, then draw the second vector from the "end" of the first vector, after that, make sure to draw arrows on the ends of the vectors so you know which direction the vectors are going and what next to do.

There are two basic ways that you can use trigonometry to find the resultant of two vectors, and which method you need depends on whether or not the vectors form a right angle or not. In this situation, they do form a right angle, so you can use the "Pythagorean Theorem" to find the length of the hypotenuse of the triangle, which will give you the magnitude of the resultant of the vector.

you have to remember that all vectors have both a magnitude and a direction, that's what makes them vectors! if you just found the magnitude of the resultant, but you also need to find its direction. To find it, you can use the trigonometric function tangent, which is equal to the side opposite the angle (20 m) divided by the side adjacent to the angle (50 m) in the triangle.

if the vectors do not form right angle, then in this solution its different when the vectors do not form a right angle! e.g: like two forces are applied to the same object. A 30 N force is applied horizontally, and a 45 N force is applied at an angle of 40 degrees to the horizontal. What is the total force applied to the object? thats when you have to calculate using the vectors formula. :) 

Hope you understand! feel free to ask questions and I am happy to help! :) 

Best Regards! 

~Jessica.