To find the unit vector in the direction of a given vector \(\mathbf{v}\), we first need to compute the magnitude of \(\mathbf{v}\) and then divide \(\mathbf{v}\) by its magnitude. The magnitude of a vector \(\mathbf{v} = ai + bj + ck\) is given by the formula:
\[ |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \]
After finding the magnitude, the unit vector \(\mathbf{u}\) in the direction of \(\mathbf{v}\) is calculated as:
\[ \mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|} \]
a. For the vector \(-3i + 7j\):
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Identify the components: \[ \mathbf{v} = -3i + 7j \quad \Rightarrow \quad a = -3, , b = 7, , c = 0 \]
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Calculate the magnitude: \[ |\mathbf{v}| = \sqrt{(-3)^2 + 7^2 + 0^2} = \sqrt{9 + 49} = \sqrt{58} \]
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Find the unit vector: \[ \mathbf{u} = \frac{-3i + 7j}{\sqrt{58}} = -\frac{3}{\sqrt{58}}i + \frac{7}{\sqrt{58}}j \]
Thus, the unit vector in the direction of \(-3i + 7j\) is: \[ \mathbf{u} = -\frac{3}{\sqrt{58}}i + \frac{7}{\sqrt{58}}j \]
b. For the vector \(8i - j + 4k\):
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Identify the components: \[ \mathbf{v} = 8i - j + 4k \quad \Rightarrow \quad a = 8, , b = -1, , c = 4 \]
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Calculate the magnitude: \[ |\mathbf{v}| = \sqrt{8^2 + (-1)^2 + 4^2} = \sqrt{64 + 1 + 16} = \sqrt{81} = 9 \]
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Find the unit vector: \[ \mathbf{u} = \frac{8i - j + 4k}{9} = \frac{8}{9}i - \frac{1}{9}j + \frac{4}{9}k \]
Thus, the unit vector in the direction of \(8i - j + 4k\) is: \[ \mathbf{u} = \frac{8}{9}i - \frac{1}{9}j + \frac{4}{9}k \]
In summary:
- a. The unit vector for \(-3i + 7j\) is \(-\frac{3}{\sqrt{58}}i + \frac{7}{\sqrt{58}}j\).
- b. The unit vector for \(8i - j + 4k\) is \(\frac{8}{9}i - \frac{1}{9}j + \frac{4}{9}k\).
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