# Can a vector of magnitude zero have a nonzero component?

During baseball practice, a batter hits a very high fly ball, and then runs in a straight line and catches it. Which had the greater displacement, the player or the ball or neither? Explain.

I think it's neither but i'm not sure and i don't understand why. could someone please explain.

Also, Can a vector of magnitude zero have a nonzero component?

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HallsofIvy
Homework Helper
"displacement" is the straight line distance between the starting point and the ending point. Both batter and ball started at the same point and ended at the same point.

(Strictly speaking, "displacement" is the vector form the starting point to the ending point- but it doesn't make sense to talk about one vector being "greater" than another so I assume you really mean the magnitude of the displacement.)

ok so i understand the first question now. but i'm still confused about if a vector of magnitude zero have a nonzero component?

A vector CAN NOT have a non-zero component and a zero magnitude.
Example in a 3D space:
Consider a vector $$\vec{V}=(v_1,v_2,v_3)$$.
By definition its magnitude is:
$$\|\vec{V}\|= \sqrt{\vec{V} \cdot \vec{V}}$$
That is $$\|\vec{V}\|=\sqrt{v_1^2+v_2^2+v_3^2}$$
Because the square of any number is always positive, it is clear that:
1- $$v_1 \neq 0$$ or $$v_2 \neq 0$$ or $$v_3 \neq 0$$ leads to $$\|\vec{V}\|>0$$,
2- $$\|\vec{V}\|=0$$ implies that $$v_1 = v_2 = v_3 = 0$$

Does it make it any clearer?

Of course you presume that the ball is struck and caught at the same height.

HallsofIvy