Given two lines in 3D space, defined by points A(1,2,3), B(4,5,6) for the first line and C(0,-1,2), D(3,0,5) for the second line, find the angle between these lines.
(a) Determine the direction vectors for each line using the given points.
Question
Determine the direction vectors \(\vec{AB}\) and \(\vec{CD}\) for the two lines. Show your working clearly.
Hint
Hint
Hint
To find the direction vector \(\vec{AB}\), subtract the coordinates of point A from point B. Similarly, for \(\vec{CD}\), subtract the coordinates of point C from point D.
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Complete Answer
\(\vec{AB} = (3, 3, 3)\) and \(\vec{CD} = (3, 1, 3)\)
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Question
Use the dot product formula to find an expression for the cosine of the angle \(\theta\) between the direction vectors \(\vec{AB}\) and \(\vec{CD}\). Simplify your expression.
Hint
Hint
Hint
The dot product formula is given by \(\vec{u} \cdot \vec{v} = \|\vec{u}\| \|\vec{v}\| \cos\theta\), where \(\|\vec{u}\|\) and \(\|\vec{v}\|\) are the magnitudes of vectors \(\vec{u}\) and \(\vec{v}\) respectively.
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- Formula Booklet
Complete Answer
\(\cos\theta = \frac{7}{\sqrt{57}}\)
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Question
Calculate the angle \(\theta\) between the two lines in degrees. Give your answer to two decimal places.
Hint
Hint
Hint
Use the inverse cosine function (\(\cos^{-1}\) or arccos) to find the angle \(\theta\) from the value of \(\cos\theta\) obtained in part (b). Ensure your calculator is in degree mode.
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- Formula Booklet
Complete Answer
\(\theta \approx 37.48^{\circ}\)
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