To find the magnitude of a vector given its components, we use the Pythagorean theorem. The magnitude [tex]\( \| \mathbf{v} \| \)[/tex] of a vector with components [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is given by:
[tex]\[ \| \mathbf{v} \| = \sqrt{x^2 + y^2} \][/tex]
For the given vector:
- The x-component ([tex]\( x \)[/tex]) is 8.72 meters.
- The y-component ([tex]\( y \)[/tex]) is 9.22 meters.
Step-by-step solution:
1. Square both the x-component and the y-component.
[tex]\[ x^2 = 8.72^2 = 76.0384 \][/tex]
[tex]\[ y^2 = 9.22^2 = 85.0084 \][/tex]
2. Add these squared values together.
[tex]\[ x^2 + y^2 = 76.0384 + 85.0084 = 161.0468 \][/tex]
3. Take the square root of the sum to find the magnitude of the vector.
[tex]\[ \| \mathbf{v} \| = \sqrt{161.0468} \approx 12.6904 \][/tex]
Therefore, the magnitude of the vector is approximately [tex]\( 12.6904 \)[/tex] meters.
