To find the sine of the angle between the vectors [tex]\(\vec{X} = 5 \hat{i} - 3 \hat{j} + 4 \hat{k}\)[/tex] and [tex]\(\vec{Y} = \hat{j} - \hat{k}\)[/tex], follow these steps:
1. Calculate the dot product of [tex]\(\vec{X}\)[/tex] and [tex]\(\vec{Y}\)[/tex]:
The dot product of two vectors [tex]\(\vec{X} = \begin{bmatrix} 5 \\ -3 \\ 4 \end{bmatrix}\)[/tex] and [tex]\(\vec{Y} = \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix}\)[/tex] is given by:
[tex]\[ \vec{X} \cdot \vec{Y} = (5 \cdot 0) + (-3 \cdot 1) + (4 \cdot -1) = 0 - 3 - 4 = -7 \][/tex]
2. Calculate the magnitudes of [tex]\(\vec{X}\)[/tex] and [tex]\(\vec{Y}\)[/tex]:
The magnitude of vector [tex]\(\vec{X}\)[/tex] is:
[tex]\[ |\vec{X}| = \sqrt{5^2 + (-3)^2 + 4^2} = \sqrt{25 + 9 + 16} = \sqrt{50} = 7.071 \][/tex]
The magnitude of vector [tex]\(\vec{Y}\)[/tex] is:
[tex]\[ |\vec{Y}| = \sqrt{0^2 + 1^2 + (-1)^2} = \sqrt{0 + 1 + 1} = \sqrt{2} = 1.414 \][/tex]
3. Calculate the cosine of the angle between [tex]\(\vec{X}\)[/tex] and [tex]\(\vec{Y}\)[/tex]:
The cosine of the angle [tex]\(\theta\)[/tex] between [tex]\(\vec{X}\)[/tex] and [tex]\(\vec{Y}\)[/tex] is:
[tex]\[ \cos \theta = \frac{\vec{X} \cdot \vec{Y}}{|\vec{X}| |\vec{Y}|} = \frac{-7}{7.071 \cdot 1.414} = \frac{-7}{10} = -0.700 \][/tex]
4. Calculate the sine of the angle using the identity [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]:
Once we have [tex]\(\cos \theta\)[/tex], we can find [tex]\(\sin \theta\)[/tex] using:
[tex]\[ \sin^2 \theta = 1 - \cos^2 \theta \][/tex]
[tex]\[ \sin^2 \theta = 1 - (-0.700)^2 \][/tex]
[tex]\[ \sin^2 \theta = 1 - 0.49 \][/tex]
[tex]\[ \sin^2 \theta = 0.51 \][/tex]
[tex]\[ \sin \theta = \sqrt{0.51} = 0.714 \][/tex]
Therefore, the sine of the angle between [tex]\(\vec{X}\)[/tex] and [tex]\(\vec{Y}\)[/tex] is approximately [tex]\(0.714\)[/tex].
