Possible Dot Product

Geometry Level 1

Given two vectors $\vec{u}$ and $\vec{v}$ such that $\|\vec{u}\| = 5$ and $\|\vec{v}\| = 8$ , what is the positive difference between the largest and smallest possible values of $\vec{u}\cdot\vec{v}?$

The answer is 80.

1 solution

Chung Kevin
Oct 1, 2015

Because $\vec{u}\cdot\vec{v} = \|\vec{u}\|\|\vec{v}\|\cos(\theta)$ , the largest and smallest values of $\vec{u}\cdot\vec{v}$ are $\|\vec{u}\|\|\vec{v}\|$ and $-\|\vec{u}\|\|\vec{v}\|$ and they occur at $\theta = 0$ and $\theta = \pi$ , respectively.

Therefore, the difference of the largest and smallest values is $40 - (-40) = \boxed{80}$

The question should probably read "difference between the largest and smallest POSSIBLE values of u v," since if u,v are defined, u v will only have one value.

David Ortiz - 3 years, 2 months ago

I edited that in.

Chung Kevin - 3 years, 2 months ago

I don’t have enough idea of eni of that question

Spiro Jorgo - 1 year, 10 months ago

I don’t have eni idea wat is possible question is going to be the next question

Spiro Jorgo - 1 year, 10 months ago

Isn't the null vector the smallest (in terms of magnitude)?

Shubhrajit Sadhukhan - 1 month, 1 week ago

Possible Dot Product

The answer is 80.

1 solution

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