- Recall that perpendicular vectors satisfy \(\mathbf{u} \cdot \mathbf{v} = 0\).
- Use the magnitude formula for the sum of vectors, \(\|\mathbf{u} + \mathbf{v}\| = \sqrt{\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2 + 2(\mathbf{u} \cdot \mathbf{v})}\), to show \(\|\mathbf{u} + \mathbf{v}\| = \sqrt{\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2}\).
- Compare \(\|\mathbf{u} + \mathbf{v}\|\) with \(\|\mathbf{u}\| + \|\mathbf{v}\|\) to establish the inequality \(\|\mathbf{u} + \mathbf{v}\| < \|\mathbf{u}\| + \|\mathbf{v}\|\).
- Conclude that the true relationships are \(\|\mathbf{u} + \mathbf{v}\| = \sqrt{\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2}\) and \(\|\mathbf{u} + \mathbf{v}\| < \|\mathbf{u}\| + \|\mathbf{v}\|\).
- \(\boxed{\|\mathbf{u} + \mathbf{v}\| = \sqrt{\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2} \text{ and } \|\mathbf{u} + \mathbf{v}\| < \|\mathbf{u}\| + \|\mathbf{v}\|}\)
Explanation
- Analyze the properties of perpendicular vectors.
Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are given to be perpendicular. The condition for two vectors to be perpendicular is that their dot product is zero, i.e., \(\mathbf{u} \cdot \mathbf{v} = 0\). We need to determine which of the given relationships regarding the magnitude of their sum \(\|\mathbf{u} + \mathbf{v}\|\) hold true. The magnitude of the sum of two vectors is determined by the dot product formula:
\[\|\mathbf{u} + \mathbf{v}\| = \sqrt{(\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} + \mathbf{v})} = \sqrt{\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2 + 2(\mathbf{u} \cdot \mathbf{v})}\]
Since the vectors are perpendicular, we can substitute \(\mathbf{u} \cdot \mathbf{v} = 0\) into this formula.
- Verify the magnitude expression.
Substituting \(\mathbf{u} \cdot \mathbf{v} = 0\) into the expression for the magnitude, we obtain:
\[\|\mathbf{u} + \mathbf{v}\| = \sqrt{\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2 + 2(0)} = \sqrt{\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2}\]
This result confirms that the second option, \(\|\mathbf{u} + \mathbf{v}\| = \sqrt{\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2}\), is true.
- Evaluate the remaining relationships.
Next, compare the magnitude of the sum to the sum of the magnitudes.
- First, check if \(\|\mathbf{u} + \mathbf{v}\| = \|\mathbf{u}\| + \|\mathbf{v}\|\). Squaring both sides:
- LHS squared: \(\|\mathbf{u} + \mathbf{v}\|^2 = \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2\)
- RHS squared: \((\|\mathbf{u}\| + \|\mathbf{v}\|)^2 = \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2 + 2\|\mathbf{u}\|\|\mathbf{v}\|\)
For non-zero vectors, \(2\|\mathbf{u}\|\|\mathbf{v}\| > 0\), so \(\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2 < (\|\mathbf{u}\| + \|\mathbf{v}\|)^2\). Thus, \(\|\mathbf{u} + \mathbf{v}\| = \|\mathbf{u}\| + \|\mathbf{v}\|\) is false.
- Second, check if \(\|\mathbf{u} + \mathbf{v}\| < \|\mathbf{u}\| + \|\mathbf{v}\|\). Based on the previous comparison, this inequality is true for non-zero vectors because \(\sqrt{\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2} < \|\mathbf{u}\| + \|\mathbf{v}\|\).
- Finally, the third option, \(\|\mathbf{u} + \mathbf{v}\| < \sqrt{\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2}\), is false because they are strictly equal.
- Final conclusion.
Based on the analysis, the two correct relationships are:
- \(\|\mathbf{u} + \mathbf{v}\| = \sqrt{\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2}\)
- \(\|\mathbf{u} + \mathbf{v}\| < \|\mathbf{u}\| + \|\mathbf{v}\|\)
The final answer is \(\boxed{\|\mathbf{u} + \mathbf{v}\| = \sqrt{\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2} \text{ and } \|\mathbf{u} + \mathbf{v}\| < \|\mathbf{u}\| + \|\mathbf{v}\|}\).
Examples
Imagine you are navigating a drone, and it travels with two perpendicular displacement vectors, \(\mathbf{u}\) (moving due North) and \(\mathbf{v}\) (moving due East). To find the drone's total distance from the starting point, you calculate the magnitude of the sum of these two displacement vectors. Since the movements are perpendicular, you can find the resulting displacement using the Pythagorean theorem, which corresponds to \(\|\mathbf{u} + \mathbf{v}\| = \sqrt{\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2}\). This calculation perfectly demonstrates why this magnitude formula is essential for finding resultant distances in two-dimensional space.