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If $\overrightarrow{x} = \langle c, c, c, 0\rangle$ for any $c \in \mathbb{R}$ what is the geometric representation of $\overrightarrow{x}$?

Im struggling to visualize this. Can I get some hints?

2 Answers 2

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I guess it is hard to visualize it because it is a vector on 4 dimensions. We are only familiar with up to 3 dimensions. Nonetheless, your vector has its fourth coordinate set to 0 and all others are equal amongst them. Let us think of vectors that have their starting point on the origin. We can instead try to picture the vector $x = (c, c, 0), c \in \Bbb R $ and try to generalize. Such a vector $x $, with third coordinate 0, is laying on our $xOy$ plane and is equally distant from both the x- and y-axis. The vector you ask about would be something similar, being equally distant from the x-, y- and z-axis while still "laying in some other plane" where the fourth coordinate is 0.

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    This doesn't seem to be quite correct unless you are fixing the starting point of a vector, say at the origin (position vector). For example, consider the vector $v=(1,1)$, this could be the vector $\vec{AB}$, where $A=(1,0)$ and $B=(2,1)$. But $\vec{AB}$ is not equidistant from the $x$ and $y$ axes. – 2017-01-09
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    @AnuragA I overlooked the fact that vectors can be translated from the origin. I will makean edit now; thanks. – 2017-01-09
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Your notation usage is a bit confusing.

The given expression is not a single vector. It represents a set of vectors and that set is $\{c(1,1,1,0) \, | \, c \in \mathbb{R}\}$. So this set represents a straight line through the origin along the direction of the vector $(1,1,1,0)$.