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Given vectors $\overrightarrow{AB}, \overrightarrow{DC},$ and $ \overrightarrow{BC}$, simplify $\overrightarrow{AB} − \overrightarrow{DC} + \overrightarrow{BC} $

Here's what I have done:

$\overrightarrow{AB} − \overrightarrow{DC} + \overrightarrow{BC}$

$\overrightarrow{AB} + \overrightarrow{CD} + \overrightarrow{BC}$

By Triangle law, $\overrightarrow{AB} + \overrightarrow{CD} = \overrightarrow{AD}$

Plugging back in we get:

$\overrightarrow{AD} + \overrightarrow{BC}$

Again using Triangle law: $\overrightarrow{AD} + \overrightarrow{BC} = \overrightarrow{AC}$

But, my online grader said it was wrong. Please help!

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  • $\begingroup$ Triangle law tells you that you can add vectors when the head of one vector lies on the tail of the other, i.e. $\vec{AB}+\vec{BC}=\vec{AC}$. $\endgroup$ – Julian Mejia May 15 at 23:19
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This statement

By Triangle law, $\overrightarrow{AB} + \overrightarrow{CD} = \overrightarrow{AD}$

is not right.

We can switch the order of addition:

$$\overrightarrow{AB} + \overrightarrow{CD} + \overrightarrow{BC}= \overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CD}= \overrightarrow{AD}$$

It tells us first traveling form $A$ to $B$, then $B$ to $C$, and from $C$ to $D$ is equivalent to directly travel from $A$ to $D$.

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  • $\begingroup$ Thanks so much, Siong! Much Appreciated. $\endgroup$ – Vanadis May 16 at 15:49

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