is it possible for the magnitude of a vector to be smaller than the magnitude of its components?

Answer 1

The answer can be yes or no depending on how you group your components. For example, If you group your components according to X, Y, and Z directions so that you only have 3 components (one representing all vector components in X and -X; one representing all vector components in Y and -Y; and one representing all vector components in Z and -Z), then the vector magnitude must be equal to or greater than the magnitude of the three vector components. The magnitude is simply the square root of the sum of the squares of the components. This is the same as the hypotenuse of a right triangle per the Pythagorean Theorem. This is the normal way to approach such problems and is probably the answer you want.

However, if you do not group all your components first, then you could have components in (for example) -x and X canceling out resulting in a vector magnitude less than that of the components.

Answer 2

Yes!

At best, the magnitude of a vector can only be EQUAL to the magnitude of it's components. This is only the case if all of the components are in the same direction.

If you add two vectors:

3' East
plus
2' West

The magnitude of the end vector is:

1' East.
.

Answer 3

It depends on what you mean by "component vector."

If you're talking about the vector of a single force/motion/etc, and breaking it into its x / y component vectors, then only special cases exist where the vector is equal to its components, but it can NEVER be smaller than a component. (given that you're saying V is a velocity vector, with Vx and Vy as its x and y component vectors).

Basically, a Vector, drawn out, with its x and y components, makes a right triangle. One rule of geometry is that in a triangle (right or otherwise) any one side MUST be longer than the other two combined. If it's longer than BOTH components, then it's definitely longer than either one of them alone.

One special case is where V = Vx and Vy = 0. (a horizontal velocity vector, where the x component is the only nonzero component). Even then, the actual vector is not longer than the component.

The other possibility is that you're talking about a resultant vector from the sum of several components. In this case, yes the resultant can, and is more often in nature than not, actually, smaller than the vectors being added together to make the resultant vector.

Take for instance, the forces on your feet when you're standing in an elevator. There's the vector of your weight pointing straight down. There's another force vector of equal magnitude pointing upward, which cancels out the force of your weight, so you're not falling down through the floor. The net force vector is zero, so you're standing still and not moving, but there are two components of this vector which are larger.

When the elevator starts moving up, there is another force vector pressing upward against your feet - the acceleration of the elevator. If you subtract the vector that is gravity's acceleration from this, then you're left with a positive vector pointing upwards, which is what causes you to accelerate upwards.

Of course, relative to the elevator, you're not moving, so there's more force being exherted downward by your body through your feet to balance out the upward acceleration, and this is why you feel heavier when it starts moving.

I hope this helps you understand vectors a little more.

Answer 4

No. Although I can not cite the Theorem which dictates this.
One way to prove this is by noting that the Length "L" of a vector is equal the the sqare root of the squares of the vectors' components. Or,

L = ||a|| = (a1^2+a2^2+a3^2)^(1/2)

The Length (also magnitude) must always be greater than or equal to the components.

Answer 5

Yes for example if the magnitude of the two components are -50 and +50 then the magnitude of the vector is 0

Answer 6

Yes. IF the two vectors are pointing opposite directions. They essentially cancel each other either in part or in whole.

Vectors have direction and magnitude.

Answer 7

No. It can be equal to the magnitude of a component
if it is only in that direction, but that's as small as it can be.

Answer 8

no, think of a right triangle and the vector is the hypotenuse. There are the x component and the y component, if x is zero then the vector is the y conponent

Answer 9

Yes.

The vector (1,1) has magnitude sqrt(2) = 1.4142..
while the magnitude of each component is 1.000.

That's just one example.

.