how is this a vector --> a = 3i + 5j - 2k ?

Question

I am picturing *one location on the x-axis, *one location on y-axis, and *one location on the z-axis. I visuallize these plotting one single point in three-dimensional space, similar to a standar x-y graph. How is this *point supposed to represent a vector????? Does the vector start from the origin and go through this point?

Answer 1

Correct. The vector points from the origin to the point.
The notation implies this.
i is a one unit vector along the positive x-axis (from origin to x=1).
j is a unit vector along the y-axis
k along the z.

Answer 2

Draw a graph like this.

|
|
____|____
/
/
/

[[edit]] Didn't show up like I wanted it. Just draw a cube and erasing all but 3 lines going in different directions on different axis, and just plot it from there.

Kinda hard to represent, the Z or depth axis (the /'s)

But start at the origin, (0,0,0) and draw a line to the point (3,5,-2) and there's your vector.

You should note that vectors have a length and direction. The point your start at is arbitrary. You can start at the origin or something like (5,5,5) and have a line to (8,10,3) by just adding your vector to the the origin.

Both of these vectors are equal even though they are in different spaces. All it takes for vectors to be equal is the same length, and being parrallel.

Hope this helps :)

Answer 3

It starts anywhere in 3-dimensional space actually. The other way of writing this vector is <3, 5, -2>, which just tells you that from wherever you are, you go three units in the x-direction, 5 units in the y-direction, and -2 in the z-directions. I struggled with understanding vectors for a while too. I'm not sure if you know how to solve for the magnitude of a vector, but in this case the magnitude is:

sqrt(3^2 + 5^2 + (-2)^2)
sqrt(9+25+4)
Sqrt(38)

So, your unit vector is Sqrt(38)*<3/Sqrt(38), 5/sqrt(38), -2/sqrt(38)>

That actually breaks a vector into 2 components: length (or magnitude), and direction. You now know that the vector is pointing in the directions of this:

<3/Sqrt(38), 5/sqrt(38), -2/sqrt(38)>

AND you know that it is the square root of 38 units long.

Yeah, vectors are challenging to learn, but very important in physics and advanced calculus.

Answer 4

It has 3 units in the x direction; 5 units in the y direction and 2 units in the negative z direction. Vectors don't have to start at the origin. You could start it out from the origin if you want to. Any other vector parallel to this vector with the same components is an equivalent vector.

Answer 5

A vector has magnitude and direction. You have correctly visualized those qualities. However, a vector has no particular location unless you specify one. It could start at the origin or any other point you choose as long as it has the same magnitude and direction.

Answer 6

It can be taken as a point, or as a vector from the origin to that point, depending on the nature of the problem.

Answer 7

imagine it like a pyramid, going from the top point to the bottom.