To calculate the magnitude of a resultant vector analytically, you typically start by determining the individual components of the vectors involved. If you have two vectors, \( \mathbf{A} \) and \( \mathbf{B} \), you can express each vector in terms of its components along the axes of a coordinate system (usually Cartesian coordinates).
For example, let's assume you have the following vectors:
- \( \mathbf{A} = A_x \hat{i} + A_y \hat{j} \) (where \( A_x \) and \( A_y \) are the components along the x and y axes, respectively)
- \( \mathbf{B} = B_x \hat{i} + B_y \hat{j} \)
Steps to calculate the resultant vector and its magnitude:
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Determine the Components of the Resultant Vector: The resultant vector \( \mathbf{R} \) is the vector sum of \( \mathbf{A} \) and \( \mathbf{B} \):
\[ \mathbf{R} = \mathbf{A} + \mathbf{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j} \]
Let’s denote the components of the resultant vector as:
- \( R_x = A_x + B_x \)
- \( R_y = A_y + B_y \)
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Calculate the Magnitude of the Resultant Vector: The magnitude \( R \) of the resultant vector \( \mathbf{R} \) can be calculated using the Pythagorean theorem:
\[ R = \sqrt{R_x^2 + R_y^2} \]
Substituting in the expressions for \( R_x \) and \( R_y \):
\[ R = \sqrt{(A_x + B_x)^2 + (A_y + B_y)^2} \]
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Generalization: This method can be extended to vectors in three dimensions by adding a z-component:
- \( \mathbf{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} \)
- \( \mathbf{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} \)
Then the resultant vector \( \mathbf{R} \) in three dimensions is:
\[ \mathbf{R} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j} + (A_z + B_z) \hat{k} \]
The magnitude then becomes:
\[ R = \sqrt{(A_x + B_x)^2 + (A_y + B_y)^2 + (A_z + B_z)^2} \]
By following these steps, you can analytically determine the magnitude of the resultant vector from a set of given vector components.