How to Find the Sine of the Angle Between Two Vectors
When working with vectors in mathematics and physics, it is often important to determine the angle between them. One way to find the sine of this angle is through the use of the cross product and the magnitudes of the vectors involved. This article will provide a detailed step-by-step guide on how to calculate the sine of the angle between two vectors using a common example.
Introduction
This method involves understanding vector operations such as the cross product and dot product. The cross product gives a vector result that is perpendicular to the two input vectors, while the dot product yields a scalar value related to the cosine of the angle between the vectors. By combining these operations, we can determine the sine of the angle as well.
Example: Finding the Sine of the Angle Between Two Vectors
Let's consider two vectors mathbf{a} mathbf{i} - 2mathbf{j} 3mathbf{k} and mathbf{b} 3mathbf{i} - 2mathbf{j} mathbf{k}. We will now go through the process to find the sine of the angle between them.
Step 1: Calculate the Cross Product mathbf{a} times mathbf{b}
The cross product of two vectors results in a vector that is orthogonal to both.
Given vectors:
[mathbf{a} mathbf{i} - 2mathbf{j} 3mathbf{k}]
[mathbf{b} 3mathbf{i} - 2mathbf{j} mathbf{k}]
The cross product can be calculated using the determinant of a matrix:
[mathbf{a} times mathbf{b} begin{vmatrix} mathbf{i} mathbf{j} mathbf{k} 1 -2 3 3 -2 1 end{vmatrix}]
Expanding the determinant:
[mathbf{a} times mathbf{b} mathbf{i} begin{vmatrix} -2 3 -2 1 end{vmatrix} - mathbf{j} begin{vmatrix} 1 3 3 1 end{vmatrix} mathbf{k} begin{vmatrix} 1 -2 3 -2 end{vmatrix}]
Calculating the 2x2 determinants:
[begin{vmatrix} -2 3 -2 1 end{vmatrix} (-2)(1) - (3)(-2) -2 6 4]
[begin{vmatrix} 1 3 3 1 end{vmatrix} (1)(1) - (3)(3) 1 - 9 -8]
[begin{vmatrix} 1 -2 3 -2 end{vmatrix} (1)(-2) - (-2)(3) -2 6 4]
Substituting these values back, we get:
[mathbf{a} times mathbf{b} 4mathbf{i} - 8mathbf{j} 4mathbf{k}]
Simplifying, we have:
[mathbf{a} times mathbf{b} 4mathbf{i} - 8mathbf{j} 4mathbf{k}]
The magnitude of this vector is calculated as follows:
[||mathbf{a} times mathbf{b}|| sqrt{4^2 (-8)^2 4^2} sqrt{16 64 16} sqrt{96} 4sqrt{6}]
Step 2: Calculate the Magnitudes of mathbf{a} and mathbf{b}
The magnitude of vector mathbf{a} is:
[||mathbf{a}|| sqrt{1^2 (-2)^2 3^2} sqrt{1 4 9} sqrt{14}]
The magnitude of vector mathbf{b} is:
[||mathbf{b}|| sqrt{3^2 (-2)^2 1^2} sqrt{9 4 1} sqrt{14}]
Step 3: Calculate sin theta
The sine of the angle between two vectors can be found using the formula:
[sin theta frac{||mathbf{a} times mathbf{b}||}{||mathbf{a}|| cdot ||mathbf{b}||}]
Substituting the values we calculated:
[sin theta frac{4sqrt{6}}{sqrt{14} cdot sqrt{14}} frac{4sqrt{6}}{14} frac{2sqrt{6}}{7}]
Thus, the sine of the angle between the two vectors is:
[sin theta frac{2sqrt{6}}{7}]
Conclusion
This article has provided a detailed explanation of how to find the sine of the angle between two vectors using the cross product and vector magnitudes. Understanding these concepts is crucial for advanced studies in mathematics and physics, particularly in fields that deal with vector analysis.
Frequent Misconceptions
Misconception 1: The cross product is always a scalar value.
Clarification: The cross product is a vector perpendicular to the plane containing the original vectors.
Misconception 2: The sine of the angle is always positive.
Clarification: The sign of the sine function depends on the positioning of the vectors in the coordinate system, but the absolute value is what is typically used in vector problems.