If A(1, 2, 1),B(2, 3, 2),C(2, 1, 3),and D(3, | vedclass

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(A) First,find the direction vectors of lines $AB$ and $CD$. The vector $\vec{AB} = (2-1, 3-2, 2-1) = (1, 1, 1)$. The vector $\vec{CD} = (3-2, 2-1, 4-

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$\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$

Vedclass · Answer

(A) First,find the direction vectors of lines $AB$ and $CD$. The vector $\vec{AB} = (2-1, 3-2, 2-1) = (1, 1, 1)$. The vector $\vec{CD} = (3-2, 2-1, 4-3) = (1, 1, 1)$. Since the direction vectors are identical,the lines are parallel. Now,check if the lines are the same by verifying if point $C(2, 1, 3)$ lies on line $AB$. The equation of line $AB$ is $\vec{r} = (1, 2, 1) + t(1, 1, 1) = (1+t, 2+t, 1+t)$. For $C$ to be on $AB$,$(1+t, 2+t, 1+t) = (2, 1, 3)$. This gives $1+t=2 \implies t=1$,$2+t=1 \implies t=-1$,and $1+t=3 \implies t=2$. Since $t$ is not consistent,$C$ does not lie on $AB$. Thus,the lines are parallel but distinct. Therefore,$\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$.

If $A(1, 2, 1)$,$B(2, 3, 2)$,$C(2, 1, 3)$,and $D(3, 2, 4)$,then which of the following is true?

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