# Given vectors AB, BC, and BC Simplify the following expression.

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: $$\overrightarrow{AD} + \overrightarrow{BC} = \overrightarrow{AC}$$

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

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 $$D$$.

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