8.57 m
Explanation:
To solve the problem, we have to decompose the two vectors along the two directions first:
Vector A:
- x component: Ax = +9.66 m
- y compoment: Ay = 0 (the vector lies along the x-axis)
Vector B:
- x component: ![B_x = -(12.0 m) cos 45^{\circ}=-8.49 m](/tpl/images/0449/9044/927ac.png)
- y component: ![B_y = (12.0 m) sin 45^{\circ}=8.49 m](/tpl/images/0449/9044/56ae5.png)
So now we can find the sum of the two vectors by adding the components along each axis:
![R_x = A_x + B_x = 9.66 m - 8.49 m = 1.17 m](/tpl/images/0449/9044/e53f8.png)
![R_y = A_y + B_y = 0 + 8.49 m = 8.49 m](/tpl/images/0449/9044/bc0c7.png)
And the magnitude of the sum is given by Pythagorean theorem:
![R=\sqrt{R_x^2+R_y^2}=\sqrt{(1.17 m)^2+(8.49 m)^2}=8.57 m](/tpl/images/0449/9044/76b5b.png)