Can Vectors Have Zero Magnitude with Nonzero Components?

[kraaaaamos]

Vectors and Magnitude - Theoretical -- PLEASE HELP!

Homework Statement

1. a. Can a vector have a component equal to zero and still have nonzero magnitude? Explain.
b. Can a vector have zero magnitude if one of its components is nonzero? Explain.

Homework Equations

During lecture, the prof. mentioned that magnitudes are always positive.
There is nothing in our textbook that discusses magnitues equaling zero.

Plus, I have no idea what the question is asking.
: (

The Attempt at a Solution

I went to look through other physics websites on the net. And from what I understand . . .
If a vector has nonzero components, it cannot have a magnitude of a vector because the very fact that it has a nonzero component, already implies a nonzero magnitude.

If a vector has a magnitude of nonzero, then the magnitude must come from nonzero components of a vector?

I don't know if that's right . . . and if the explanation is sufficient? = S

 

[JoAuSc]

Vectors can have zero magnitude. They just can't have negative magnitude.

 

[kraaaaamos]

I looked at the question again.

a) Yes. It can have a Y-component of zero and a non-zero x-component, which will equal to a nonzero magnitude. Therefore, a vector can have zero component, but still have a nonzero magnitude.

b) I'm not sure. Is the magnitude zero if someone travels 6m north, then 6m south?

Because in that case, it would be possible to have zero magnitude even if you have nonzero components, IF the nonzero components cancel each other out (like in the case where someone travels 6m north and then 6m south) . I don't know if that makes sense.

 

[JoAuSc]

You would need two vectors to describe the path of someone walking north, then south. Once you add the two vectors, you're left with one vector, and you can determine the north-south component, the east-west component, and the up-down component (for 3D space). You can't have two north-south components for the same vector.

 

[kraaaaamos]

I have no idea what the answer is. Can you give me a hint for these questions:

1. a. Can a vector have a component equal to zero and still have nonzero magnitude? Explain.
b. Can a vector have zero magnitude if one of its components is nonzero? Explain.

 

[JoAuSc]

You are correct on part a. For part b, you need a better understanding of what the components of a vector are. To put it simply, if you have a vector on a plane, its length along the x-axis is one component and its length along the y-axis is the other component. Now, let's say the length along the x-axis is 1.0 . Can the vector possibly have zero length?

Here's another way to think about it. If x, y, and z are the components of a vector, its length r is

r = sqrt(x^2 + y^2 + z^2)

because of the Pythagorean theorem. If r = 0, what are the possible values of x, y, and z?