The Vector A is 3cm long and points to the right. The vector B is 6cm long and makes an angle of 130 degrees with the horizontal (take 0 degrees to be to the right). The vector C= A+B. What is the lenght and direction of C?
A = (3,0)
B = (6 cos130,6 sin130)
now add the x- and y-components to get C.
Then convert to polar form as usual.
To find the length and direction of vector C, we first need to break down vectors A and B into their respective horizontal and vertical components.
For vector A, since it points to the right, the horizontal component is equal to its magnitude, which is 3 cm, and the vertical component is zero.
For vector B, we can find the horizontal and vertical components using trigonometry. The horizontal component (Bx) is given by B * cos(130°), and the vertical component (By) is given by B * sin(130°).
Let's calculate these components:
Bx = 6 cm * cos(130°) ≈ -3.39 cm (negative because it points to the left)
By = 6 cm * sin(130°) ≈ 5.27 cm (positive because it points upward)
Now, we can find the components of vector C by adding the corresponding components of vector A and B:
Cx = Ax + Bx = 3 cm + (-3.39 cm) ≈ -0.39 cm
Cy = Ay + By = 0 cm + 5.27 cm = 5.27 cm
The length of vector C can be found using the Pythagorean theorem:
|C| = sqrt(Cx^2 + Cy^2) ≈ sqrt((-0.39 cm)^2 + (5.27 cm)^2) ≈ 5.30 cm
Finally, we can determine the direction of vector C by calculating the angle it makes with the horizontal:
θ = arctan(Cy / Cx) ≈ arctan(5.27 cm / -0.39 cm) ≈ -86.4°
Therefore, the length of vector C is approximately 5.30 cm, and it makes an angle of -86.4° with the horizontal (counterclockwise direction).