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How to a $3D$-figure that describes all vectors making $ \text{negative dot product} $ with the vector $(1,1,1)$.

The vector $(-1,-1,-2)$ has negative do product with $(1,1,1)$ because

$(1,1,1) \cdot (-1,-1,-2)=-4<0$.

The figure is enter image description here

This is just particular case. But there are many vectors having negative dot products with $(1,1,1)$.

So the angle concept is required. If the angle is $ \pi/2 < \theta<3 \pi/2$ , then $ \ \cos \theta <0$ and the dot product will be negative as $a \cdot b=|a||b| \cos \theta$.

But how to draw all the vectors that are with $ \pi/2 < \theta<3 \pi/2$ angle with $(1,1,1)$.

Help me

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    $\begingroup$ Consider a plane $x+y+z=0$ which is normal to $(1,1,1)$. This plane separates $\mathbb{R}^3$ into two parts. One is above the plane, or $x+y+z>0$, and vectors in this part forms an acute angle with the vector $(1,1,1)$. Below the plane are the vectors which form oblique angles with $(1,1,1)$, and they all satisfies the equation $x+y+z<0$. You can also observe that if $(x,y,z)\cdot (1,1,1)<0$, then definitely $x+y+z<0$. $\endgroup$
    – J1U
    Oct 2, 2018 at 11:01
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    $\begingroup$ @J1U That deserves to be an answer instead of a comment. $\endgroup$
    – amd
    Oct 2, 2018 at 20:15
  • $\begingroup$ @amd I thought it was short enough to put in a comment. I'll add this as an answer. $\endgroup$
    – J1U
    Oct 3, 2018 at 4:46

1 Answer 1

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Consider a plane $x+y+z=0$ which is normal to $(1,1,1)$. This plane separates $\mathbb{R}^3$ into two parts. One is above the plane, or $x+y+z>0$, and vectors in this part forms an acute angle with the vector $(1,1,1)$. Below the plane are the vectors which form obtuse angles with $(1,1,1)$, and they all satisfies the equation $x+y+z<0$. You can also observe that if $(x,y,z)\cdot (1,1,1)<0$, then definitely $x+y+z<0$.

This works for any other vector. When a vector $(a,b,c)$ is given, vectors $(x,y,z)$ satisfying $ax+by+cz<0$ will form an obtuse angle with $(a,b,c)$, and thus the value of the inner product of those two vectors would be negative.

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  • $\begingroup$ An oblique angle is one whose measure is not $90^\circ$ or a multiple of $90^\circ$. An obtuse angle is one whose measure lies between $90^\circ$ and $180^\circ$. $\endgroup$ Oct 3, 2018 at 9:27
  • $\begingroup$ @N.F.Taussig Oh thanks. I am not very good at English :) $\endgroup$
    – J1U
    Oct 3, 2018 at 9:41

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