Calculate the magnitudes and sum of the given vectors.

Understand the Problem

The question is involving calculations with vectors, specifically dealing with their magnitudes and basic operations like addition. The problem indicates several steps demonstrating how to calculate the magnitudes and additions of the given vectors.

Answer

- The magnitude of vector $ a $ is approximately $3.61$, and the magnitude of vector $ b $ is approximately $4.12$. The resultant vector $ a + b $ is $ 6i + 2j $, while the magnitude of vector $ v $ is $7$, and the unit vector $ \hat{v} $ is $ \left(\frac{6}{7}, -\frac{3}{7}, \frac{2}{7}\right) $. The magnitude of the second $ a $ is approximately $7.07$, and the resultant of the second $ a + b $ is $ 5i + 3j $.

Steps to Solve

Calculate the magnitude of vector a
For vector ( a = (2, 3) ), the magnitude is calculated using the formula:
$$ |a| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.61 $$
Calculate the magnitude of vector b
For vector ( b = (4, -1) ), the magnitude is:
$$ |b| = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.12 $$
Adding vectors a and b
To add the vectors, combine their components:
$$ a + b = (2 + 4)i + (3 - 1)j = 6i + 2j $$
Calculate the components of vector v
For vector ( v = 6i - 3j + 2k ), to find its magnitude:
$$ |v| = \sqrt{6^2 + (-3)^2 + 2^2} = \sqrt{36 + 9 + 4} = \sqrt{49} = 7 $$
Calculate the unit vector of v
The unit vector ( \hat{v} ) is found by dividing vector ( v ) by its magnitude:
$$ \hat{v} = \frac{1}{7}(6i - 3j + 2k) = \left(\frac{6}{7}, -\frac{3}{7}, \frac{2}{7}\right) $$
Calculate the magnitude of vector a again
For vector ( a = 5i + 5j ):
$$ |a| = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} \approx 7.07 $$
Adding new vectors a and b
For ( a = 2i + 5j ) and ( b = 3i - 2j ), add them:
$$ a + b = (2 + 3)i + (5 - 2)j = 5i + 3j $$

The magnitude of vector ( a ) is approximately $3.61$.
The magnitude of vector ( b ) is approximately $4.12$.
The resultant vector ( a + b ) is ( 6i + 2j ).
The magnitude of vector ( v ) is $7$.
The unit vector ( \hat{v} ) is ( \left(\frac{6}{7}, -\frac{3}{7}, \frac{2}{7}\right) ).
The magnitude of vector ( a ) is approximately $7.07$ (from the second ( a )).
The resultant vector ( a + b ) is ( 5i + 3j ).

More Information

Magnitudes provide the length of the vectors, which is essential in physics and engineering for understanding force and direction.
Unit vectors indicate direction and help simplify vector calculations.

Tips

A common mistake is miscalculating the square root, especially when summing squares. Always double-check your arithmetic.
Mixing up the signs when adding or subtracting vector components can lead to incorrect results.

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