Problem 3
Question
Three points \(p_{j}=\left(x_{j}, y_{j}, z_{j}\right)\) which do not lie in a plane through the origin determine a trihedral with sides \(\overrightarrow{0 p_{1}}, \overrightarrow{\overrightarrow{b p}_{2}}, \overrightarrow{0 p_{3}}\) which has positive orientation if and only if $$ \left|\begin{array}{lll} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{array}\right|>0 $$ Using this, show that the vectors a, b, and \(\mathrm{a} \times \mathbf{b}\) form a trihedral having positive orientation. unless a and \(\mathrm{b}\) are parallel.
Step-by-Step Solution
Verified Answer
Vectors a, b, and \( \mathbf{a} \times \mathbf{b} \) form a trihedral having positive orientation, unless a and b are parallel. This is due to the determinant of the matrix formed by their coordinates being positive, unless a and b are parallel in which case the determinant and therefore the cross product will be zero.
1Step 1: Understand Orientations and Cross Product
Orientations are units of measurement for vector classification, while a trihedral is a figure in three-dimensional space formed by three plane faces. A positive orientation indicates a clockwise system, whereas a negative orientation indicates a counter-clockwise system. The cross product, denoted as \( \mathbf{a} \times \mathbf{b} \), is another vector that is perpendicular (or orthogonal) to both \( \mathbf{a} \) and \( \mathbf{b} \). This will form a third side necessary for a trihedral.
2Step 2: Formulate Condition for Positive Orientation
The given vectors form a positive orientation if and only if the determinant of the matrix formed by their coordinates is positive, as stated by the exercise condition. This condition can be written as: \[ \left|\begin{array}{lll} a_{1} & a_{2} & a_{3} \ b_{1} & b_{2} & b_{3} \ (a_{1} \times b_{1}) & (a_{2} \times b_{2}) & (a_{3} \times b_{3}) \end{array}\right| > 0 \]
3Step 3: Compute the Cross Product
Calculate \( \mathbf{a} \times \mathbf{b} \). The cross product is found by using determinants as well: \( \mathbf{a} \times \mathbf{b} = (a_{2}b_{3} - a_{3}b_{2}, a_{3}b_{1} - a_{1}b_{3}, a_{1}b_{2} - a_{2}b_{1}) \). This result corresponds to the last row of our determinant.
4Step 4: Evaluate the Determinant
Evaluate the determinant after substituting the cross product components: \[ \left|\begin{array}{lll} a_{1} & a_{2} & a_{3} \ b_{1} & b_{2} & b_{3} \ (a_{2}b_{3} - a_{3}b_{2}) & (a_{3}b_{1} - a_{1}b_{3}) & (a_{1}b_{2} - a_{2}b_{1}) \end{array}\right| \]
5Step 5: Show the Result Is Always Positive
The determinant will always be greater than 0, unless vectors a and b are parallel. In this case, the determinant will be equal to 0, because the vectors have the same or opposite direction, so the cross product will be the zero vector, hence the determinant will be 0. This shows that a, b, and \( \mathbf{a} \times \mathbf{b} \) form a trihedral having positive orientation, unless a and b are parallel.
Key Concepts
Cross ProductVector DeterminantPositive Orientation Condition
Cross Product
The cross product, often represented as \( \mathbf{a} \times \mathbf{b} \) in mathematics, is a binary operation on two vectors in three-dimensional space. It results in a new vector that is perpendicular to both of the vectors being multiplied. The magnitude of the cross product depends on the angle between the two original vectors; it reaches its maximum when they are perpendicular and equals zero when the vectors are parallel.
For vectors \( \mathbf{a} \) and \( \mathbf{b} \) with components \( (a_1, a_2, a_3) \) and \( (b_1, b_2, b_3) \) respectively, the cross product is given by the determinant of a matrix whose rows are the unit vectors i, j, k, followed by the components of \( \mathbf{a} \) and \( \mathbf{b} \) as:
\[ \mathbf{a} \times \mathbf{b} = \left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \a_1 & a_2 & a_3 \b_1 & b_2 & b_3\end{array}\right| \]
Performing the determinant calculation provides the components of the cross product vector. Understanding the cross product is crucial when addressing vector orientation, as it is the foundation for constructing a trihedral with a specified orientation as seen in the given exercise.
For vectors \( \mathbf{a} \) and \( \mathbf{b} \) with components \( (a_1, a_2, a_3) \) and \( (b_1, b_2, b_3) \) respectively, the cross product is given by the determinant of a matrix whose rows are the unit vectors i, j, k, followed by the components of \( \mathbf{a} \) and \( \mathbf{b} \) as:
\[ \mathbf{a} \times \mathbf{b} = \left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \a_1 & a_2 & a_3 \b_1 & b_2 & b_3\end{array}\right| \]
Performing the determinant calculation provides the components of the cross product vector. Understanding the cross product is crucial when addressing vector orientation, as it is the foundation for constructing a trihedral with a specified orientation as seen in the given exercise.
Vector Determinant
The vector determinant is a mathematical calculation used to find the volumetric span of three vectors in three-dimensional space. The determinant could be considered a scalar value that represents the volume of the parallelepiped formed by the vectors.
When we place the three vectors whose determinant we wish to find as rows (or columns) in a square matrix, the determinant of this matrix can be computed using a specific formula. For instance, considering vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) as rows in a matrix:\[ \left|\begin{array}{ccc}a_{1} & a_{2} & a_{3} \b_{1} & b_{2} & b_{3} \c_{1} & c_{2} & c_{3}\end{array}\right| \]
This determinant, also called a scalar triple product when \( \mathbf{c} \) is the cross product of \( \mathbf{a} \) and \( \mathbf{b} \), has special significance in physical and geometrical contexts, indicating whether the set of vectors can span a three-dimensional space and its orientation. A non-zero determinant means the vectors are linearly independent and form a trihedral with some volume.
When we place the three vectors whose determinant we wish to find as rows (or columns) in a square matrix, the determinant of this matrix can be computed using a specific formula. For instance, considering vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) as rows in a matrix:\[ \left|\begin{array}{ccc}a_{1} & a_{2} & a_{3} \b_{1} & b_{2} & b_{3} \c_{1} & c_{2} & c_{3}\end{array}\right| \]
This determinant, also called a scalar triple product when \( \mathbf{c} \) is the cross product of \( \mathbf{a} \) and \( \mathbf{b} \), has special significance in physical and geometrical contexts, indicating whether the set of vectors can span a three-dimensional space and its orientation. A non-zero determinant means the vectors are linearly independent and form a trihedral with some volume.
Positive Orientation Condition
The positive orientation condition is an essential concept in understanding the spatial arrangement of vectors in three-dimensional geometry. This condition provides a criterion for determining whether the trihedral (or 3D corner) formed by three vectors emanates outwards in a right-handed configuration, which is considered to have a positive orientation.
A trihedral defined by vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) fulfills the positive orientation condition if the determinant of the matrix composed of these vectors is greater than zero — symbolically:\[ \left|\begin{array}{ccc}a_{1} & a_{2} & a_{3} \b_{1} & b_{2} & b_{3} \c_{1} & c_{2} & c_{3}\end{array}\right| > 0 \]
For the case presented in the exercise, the cross product \( \mathbf{a} \times \mathbf{b} \) creates the third side of the trihedral. If vectors \( \mathbf{a} \) and \( \mathbf{b} \) are not parallel, the determinant of the matrix having \( \mathbf{a}, \mathbf{b}, \mathbf{a} \times \mathbf{b} \) as rows gives a positive value, confirming the positive orientation. This not only aids in visualizing complex vector orientations but is a crucial concept in fields such as physics, engineering, and computer graphics, where understanding the directionality and handedness of a system is fundamental.
A trihedral defined by vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) fulfills the positive orientation condition if the determinant of the matrix composed of these vectors is greater than zero — symbolically:\[ \left|\begin{array}{ccc}a_{1} & a_{2} & a_{3} \b_{1} & b_{2} & b_{3} \c_{1} & c_{2} & c_{3}\end{array}\right| > 0 \]
For the case presented in the exercise, the cross product \( \mathbf{a} \times \mathbf{b} \) creates the third side of the trihedral. If vectors \( \mathbf{a} \) and \( \mathbf{b} \) are not parallel, the determinant of the matrix having \( \mathbf{a}, \mathbf{b}, \mathbf{a} \times \mathbf{b} \) as rows gives a positive value, confirming the positive orientation. This not only aids in visualizing complex vector orientations but is a crucial concept in fields such as physics, engineering, and computer graphics, where understanding the directionality and handedness of a system is fundamental.
Other exercises in this chapter
Problem 2
Evaluate \(\int_{y}(y d x-x d y)\) where (a) \(\gamma\) is the closed curve \(x=t^{2}-1, y=t^{3}-t,-1 \leq t \leq 1\) (b) \(\gamma\) is the straight line from \
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Consider the differential form $$ \omega=\frac{x d x+y d y}{x^{2}+y^{2}} $$ Show that \(d \omega=0\) in the ring \(D .\) Is \(\omega\) exact in \(D ?\)
View solution Problem 4
Evaluate \(\int_{y}\left(z d x+x^{2} d y+y d z\right)\) where (a) \(\gamma\) is the straight line from \((0,0,0)\) to \((1,1,1)\). (b) \(\gamma\) is the portion
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