Answer:
Approximately [tex]3.01\; \rm m[/tex].
Explanation:
Decompose each vector into the sum of two vectors: a horizontal one (parallel to the arrow that points to the right) and a vertical one (parallel the arrow that points upwards.)
Vector [tex]\sf A[/tex] is horizontal and is at an angle of [tex]0^\circ[/tex] with the horizon.
- Horizontal component of vector [tex]\sf A[/tex]: to the right, with a length of [tex]5.00\; \rm m \cdot \cos\left(0^\circ \right) = 5.00\; \rm m[/tex].
- Vertical component of vector [tex]\sf A[/tex]: [tex]5.00\; \rm m \cdot \sin\left(0^\circ \right) = 0\; \rm m[/tex].
Vector [tex]\sf B[/tex] is at an angle of [tex]30^\circ[/tex] below the horizon.
- Horizontal component of vector [tex]\sf B[/tex]: to the right, with a length of [tex]\displaystyle 6.00\; \rm m \cdot \cos\left(30^\circ \right) = (6.00\; \rm m) \times \frac{\sqrt{3}}{2}\approx 5.19615\; \rm m[/tex].
- Vertical component of vector [tex]\sf B[/tex]: downwards, with a length of[tex]\displaystyle 6.00\; \rm m \cdot \sin\left(30^\circ \right) = 6.00\; \rm m \times \frac{1}{2} = 3.00\; \rm m[/tex].
Calculate the sum of vector [tex]\sf A[/tex] and vector [tex]\sf B[/tex].
The horizontal component of vector [tex]\sf A[/tex] and vector [tex]\sf B[/tex] are opposite to one another. Therefore, the length of the horizontal component of [tex]\sf (A + B)[/tex] would be the difference between the length of the horizontal components of vector [tex]\sf A\![/tex] and of vector [tex]\sf B\![/tex]:
[tex]\displaystyle (6.00\; \rm m) \times \frac{\sqrt{3}}{2} - 5.00\; \rm m \approx 0.196152\; \rm m[/tex].
The length of the vertical component of vector [tex]\sf A[/tex] is [tex]0\; \rm m[/tex]. Therefore, the length of the vertical component of [tex]\sf (A + B)[/tex] would be equal to the length of the vertical component of vector [tex]\sf B[/tex], [tex]\displaystyle 6.00\; \rm m \times \frac{1}{2} = 3.00\; \rm m[/tex].
Therefore, the length of the horizontal and vertical component of [tex]\sf (A + B)[/tex] are approximately [tex]0.196152\; \rm m[/tex] and [tex]3.00\; \rm m[/tex], respectively. The length of vector [tex]\sf (A + B)\![/tex] would be approximately:
[tex]\displaystyle \sqrt{(0.196152\; \rm m)^{2} + (3.00\; \rm m)^{2}} \approx 3.01\; \rm m[/tex].
