Given that \(r = 3i + 4j\) and \(t = -5i + 12j\), find the acute angle between them.
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Correct Answer: Option C
Explanation:
\(\overrightarrow{r} . \overrightarrow{t} = |\overrightarrow{r}||\overrightarrow{t}|\cos \theta\)
\(\overrightarrow{r} . \overrightarrow{t} = (3i + 4j) . (-5i + 12j) = -15 + 48 = 33\)
\(|\overrightarrow{r}| = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5\)
\(|\overrightarrow{t}| = \sqrt{(-5)^{2} + 12^{2}| = \sqrt{169} = 13\)
\(\cos \theta = \frac{\overrightarrow{r} . \overrightarrow{t}}{|\overrightarrow{r}||\overrightarrow{t}|}\)
\(\cos \theta = \frac{33}{5 \times 13} = \frac{33}{65}\)
\(\theta = \cos^{-1} {\frac{33}{65}} \approxeq 59.5°\)
\(\overrightarrow{r} . \overrightarrow{t} = |\overrightarrow{r}||\overrightarrow{t}|\cos \theta\)
\(\overrightarrow{r} . \overrightarrow{t} = (3i + 4j) . (-5i + 12j) = -15 + 48 = 33\)
\(|\overrightarrow{r}| = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5\)
\(|\overrightarrow{t}| = \sqrt{(-5)^{2} + 12^{2}| = \sqrt{169} = 13\)
\(\cos \theta = \frac{\overrightarrow{r} . \overrightarrow{t}}{|\overrightarrow{r}||\overrightarrow{t}|}\)
\(\cos \theta = \frac{33}{5 \times 13} = \frac{33}{65}\)
\(\theta = \cos^{-1} {\frac{33}{65}} \approxeq 59.5°\)