I am just beginning to learn about three dimensional vectors and I am not really sure how to go about this problem. I set r1=r2 but I'm not quite sure where to go from here. Any help would be appreciated. Thanks!
2026-04-12 07:32:35.1775979155
Vectors in the Third Dimension
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The key fact is that if the lines intersect at a point $P$, then $P=r_1(s)=r_2(t)$ for some values of $s$ and $t$. (It doesn't have to be the same value of the parameter $t$ for each line; that's why I've changed the variable to $s$ for the first line.)
Therefore, you need $4s+9=-3t-2$ (from the first component) and $8=-3t-1$ (from the last component). So $t=-3$ (from the second equation), and hence $s=-0.5$. If you plug in $t=-3$ to the second line $r_2$, we get the point $(7,8,8)$. Plugging in $s=-0.5$ to the first line $r_1$ gives the point $(7,8,8)$. This confirms that $P=(7,8,8)$ is the point of intersection.