Consider the following scenario in 3D space:
A plane \(\Pi\) passes through the point A(1, 2, 3) and is parallel to the vectors \(u = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}\) and \(v = \begin{pmatrix} 1 \\ 1 \\ -2 \end{pmatrix}\).
A line L is given by its parametric equation: \(r = \begin{pmatrix} 4 \\ 0 \\ -1 \end{pmatrix} + t\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\), where \(t \in \mathbb{R}\).
Additionally, consider the point P(0, -2, 5).
Answer the following parts:
Question
Given that a plane \(\Pi\) passes through A(1, 2, 3) and is parallel to \(u = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}\) and \(v = \begin{pmatrix} 1 \\ 1 \\ -2 \end{pmatrix}\), determine the scalar equation of the plane \(\Pi\).
Hint
Hint
Hint
First, calculate the normal vector to the plane using the cross product of the direction vectors. Then, use the point-normal form to find the scalar equation.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
The scalar equation of the plane \(\Pi\) is: \(x + 5y + 3z – 20 = 0\).
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Answer Explanation
Question
Calculate the perpendicular distance from the point P(0, -2, 5) to the plane \(\Pi\) with equation \(x+5y+3z-20=0\).
Hint
Hint
Hint
Use the formula for the perpendicular distance from a point to a plane.
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Answer
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Explanation
- Formula Booklet
Complete Answer
The perpendicular distance from point P to the plane \(\Pi\) is \(\frac{15}{\sqrt{35}} = \frac{3\sqrt{35}}{7}\) units, approximately 2.54 units.
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Answer Explanation
Answer Explanation
Question
Calculate the acute angle between the line L, with direction vector \(d = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\), and the plane \(\Pi\) with equation \(x+5y+3z-20=0\).
Hint
Hint
Hint
Find the angle between the direction vector of the line and the normal vector of the plane, and then use the complementary angle to find the angle between the line and the plane.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
The acute angle between the line L and the plane \(\Pi\) is approximately \(25.5^\circ\).
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Answer Explanation
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