Equal But Opposite

Geometry Level 1

Suppose two vectors $\vec{u}$ and $\vec{v}$ have the same magnitude but have opposite directions. What is $\vec{u} + \vec{v}$ ?

\vec{0}

\vec{v} - \vec{u}

0

\vec{u} - \vec{v}

3 solutions

Chung Kevin
Sep 30, 2015

If $\vec{u}$ has the same magnitude as $\vec{v}$ but is in the opposite direction $\vec{v}$ , then $\vec{u} = -\vec{v}$ .

Therefore, $\vec{u} + \vec{v} = \vec{0}$ .

Why would their sum be a vector? Shouldn't it be zero as a scalar quantity?

Andrew Tawfeek - 5 years, 8 months ago

No, not at all. It's the zero vector.

A vector is an element of a vector space.

Because of this, it must satisfy the vector space axioms. It needs to be closed under vector addition. In other words, a vector plus a vector must always give another vector.

Isaac Buckley - 5 years, 8 months ago

I would rather label it a matter of notation. I mean, who cares about how the symbol of the neutral element looks like? (though I guess in school, one might do care, so in regards of perception your answer is best, it explains all there is to know) . I imagine it also rather as an object with special properties than a mere vector.. so I didn't see much of a different between $0$ and $\vec{0}$ But I think there isn't much to gain by thinking about this any further.

Alisa Meier - 5 years, 8 months ago

Shah Faisal
Jul 6, 2017

Both of these vectors have the same magnitude but as they are antiparallel so they will cancel the effect of each other.so their net effect will be zero.

Sebastian Thazhath
Aug 3, 2019

Both vectors are of equal magnitude ( length ) but they are opposite in direction so it would look like this

So we can reverse these vectors so that the base starts from where the point ( arrowhead ) of the smallest vector was. The distance between where the two new points overlap is the answer to the sum. Like this.

here there is no overlap so the sum is 0 but to show it is a vector you must place a line on top of the 0. However, if one of the vectors was bigger than the other it would look like this

Then we flip the vectors so that the bases start at the same distance from the axis as the old point of the smaller vector. like this

the distance of overlap would be the answer to the sum

Equal But Opposite

3 solutions

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