| 8.2 |
Addition and Subtraction of Vectors
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| \(\blacksquare\) A resultant vector is the combination of two or more single vectors. |
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| Parallel vectors |
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(a) The addition of two parallel vectors \(\utilde{a}\) and \(\utilde{b}\) can be combined to produce a resultant vector represented as \(\utilde{a}+\utilde{b}\).
For example,
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(b) The subtraction of vector \(\utilde{b}\) from vector \(\utilde{a}\) is the sum of vector \(\utilde{b}\) and negative vector \(\utilde{b}\), that is
\(\utilde{a}-\utilde{b}=\utilde{a}+(-\utilde{b})\).
For example,
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| \(\blacksquare\) The resultant vector can be obtained using |
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| triangle law |
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It is given that
\(\begin{aligned} \overrightarrow{CD}&=3\utilde{r}, \\\\ \overrightarrow{EF}&=5\utilde{r} ,\\\\ \overrightarrow{FG}&=\utilde{r}. \end{aligned}\)
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If \(|\utilde{r}|=3 \text{ units}\), find the magnitude of the following expression:
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| \(\begin{aligned} 2\overrightarrow{CD}+\overrightarrow{EF}-3\overrightarrow{FG} \end{aligned}\) |
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| Solution: |
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Express the expression in term of \(\utilde{r}\).
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\(\begin{aligned} &2\overrightarrow{CD}+\overrightarrow{EF}-3\overrightarrow{FG} \\\\ &=2(3\utilde{r})+5 \utilde{r}-3(\utilde{r}) \\\\ &=6\utilde{r}+5 \utilde{r}-3\utilde{r}. \end{aligned}\)
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Since \(|\utilde{r}|=3 \text{ units}\),
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| hence, |
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\(\begin{aligned} &|6\utilde{r}+5 \utilde{r}-3\utilde{r}|\\\\ &=|8\utilde{r}|\\\\&=8|\utilde{r}|\\\\ &=8(3) \\\\ &=24 \text{ units}. \end{aligned}\)
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Two forces \(\bold{F}_1=50 \text{ N}\) and \(\bold{F}_2=30 \text{ N}\) are acting on a moving object.
Determine the magnitude of the force and the direction of the movement of the object if
\(\bold{F}_1\) and \(\bold{F}_2\) are in the opposite direction.
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| Solution: |
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If \(\bold{F}_1\) and \(\bold{F}_2\) are in the opposite direction,
then
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Magnitude:
\(\begin{aligned} \bold{F}_1+(-\bold{F}_2)&=50+(-30)\\\\&=50-30 \\\\ &=20\text{ N}. \end{aligned}\)
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The magnitude of the force acted on the object is \(20\text{ N}\) in the same dircetion as \(\bold{F}_1\).
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