If a, beta be the direction cosines of a vector with respect to X and Y-axes and is its components, find the magnitude of the vector.
Understand the Problem
The question is asking us to find the magnitude of a vector given its direction cosines with respect to the X and Y axes. This involves using the relationship between direction cosines and the vector components to calculate the magnitude.
Answer
The magnitude of the vector is given by $|\mathbf{v}| = \sqrt{l^2 + m^2 + n^2}$.
Answer for screen readers
The magnitude of the vector is given by: $$ |\mathbf{v}| = \sqrt{l^2 + m^2 + n^2} $$
Steps to Solve
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Identify direction cosines Given direction cosines typically represented as $l$, $m$, and $n$ correspond to the angles the vector makes with the X, Y, and Z axes respectively.
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Use the formula for magnitude The magnitude of a vector $|\mathbf{v}|$ can be calculated using the formula: $$ |\mathbf{v}| = \sqrt{l^2 + m^2 + n^2} $$ where $l$, $m$, and $n$ are the direction cosines of the vector.
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Input the direction cosines Substitute the values of the direction cosines into the magnitude formula.
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Calculate the magnitude Perform the calculations step by step, first squaring each direction cosine, then summing them, and finally taking the square root.
The magnitude of the vector is given by: $$ |\mathbf{v}| = \sqrt{l^2 + m^2 + n^2} $$
More Information
The final value of the magnitude depends on the specific values of the direction cosines. Direction cosines are very useful in fields like physics and engineering to understand vector directions without needing to know their lengths.
Tips
- Forgetting to square the direction cosines before summing them.
- Not taking the square root at the end, leading to an incorrect magnitude.
- Confusing direction cosines with direction vectors; they are not the same.
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