# all vectors making $\text{negative dot product}$ with the vector $(1,1,1)$.

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

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

• 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 $x+y+z<0$.
– J1U
Oct 2, 2018 at 11:01
• @J1U That deserves to be an answer instead of a comment.
– amd
Oct 2, 2018 at 20:15
• @amd I thought it was short enough to put in a comment. I'll add this as an answer.
– J1U
Oct 3, 2018 at 4:46

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.
• 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 $180^\circ$. Oct 3, 2018 at 9:27