What does $x= (0,0,0)$+$\lambda(-1,-1,-1)$ mean?

Question

What does $x= (0,0,0)+\lambda(-1,-1,-1)$ mean? does it mean that the planes cross at the origin? Then what does $(-1,-1,-1)$ stand for?

Answer 1

$$x=(x_1,x_2,x_3)$$ is a point in the space which belong to the line containing the origin $(0,0,0)$ and whose vector director is $(-1,-1,-1)$ or $(1,1,1)$.

You can also see it as an intersection of two planes

$$x_1=x_2$$ and $$x_1=x_3$$

Answer 2

It is the parametric equation of a line which in general, also in $\mathbb{R^n}$, is in the form

$$P(t)=P_0+t\cdot v$$

where

• $P_0=P(0)\in \mathbb{R^n}$ is a reference point on the line

• $v\in \mathbb{R^n}$ is the direction vector

• $t\in \mathbb{R}$ is the parameter

In that case the given expression represents then a line in $\mathbb{R^3}$ passing through the origin and with direction vector $(-1,-1,-1)$.

Note that we can use any point on the line as reference point and any vector parallel to $(-1,-1,-1)$ as direction vector to represent the same line, that is for example

$$P(t)=(2,2,2)+t(3,3,3)$$