Lines in Grassmannian

Question

Let $\{x_1,x_2,x_3,x_4,x_5,x_6\}$ be a basis of $\mathbb C^6$. Let $V \in \mathrm{Gr}(3,\mathbb C^6)$ be such that $$V=\left<\sum_{i=1}^6a_ix_i, \sum_{i=1}^6b_ix_i, \sum_{i=1}^6c_ix_i\right>$$ and $W \in\mathrm{Gr}(3, \mathbb C^6)$ be such that $$W=\left<\sum_{i=1}^3a_ix_i-\sum_{i=4}^6a_ix_i, \sum_{i=1}^3b_ix_i-\sum_{i=4}^6b_ix_i, \sum_{i=1}^3c_ix_i -\sum_{i=4}^6c_ix_i\right>.$$ Then $V$ and $W$ represent two points in $\mathbb P(\bigwedge^3\mathbb C^6)$ denote them by $[V]$ and $[W]$. My guess is that the line joining these two points in $\mathbb P(\bigwedge^3\mathbb C^6)$ intersects $\mathrm{Gr}(3, \mathbb C^6)$ exactly at two points namely at $[V]$ and $[W]$. Is there a way to prove my claim ? Thanks.