Vector ~A has a magnitude of 7 and points in the positive x-direction. Vector ~B has a magnitude of 21 and makes an angle of 30â—¦ with the positive x-axis.
What is the magnitude of ~A − ~B ?
X = 7 + 21Cos(30) = 7 + 18.19 = 25.19,
Y = 21Sin(30) = 10.5,
Z^2 = X^2 + Y^2 = (25.19)^2 + (10.5)^2 = 634.54 + 110.25 = 744.8,
Z = sqrt(744.8) = 27.3 = Magnitude.
To find the magnitude of the difference between two vectors ~A and ~B, we need to subtract the components of ~B from ~A and then calculate the magnitude of the resulting vector.
Given that ~A has a magnitude of 7 and points in the positive x-direction, we can represent it as ~A = 7i, where i is the unit vector in the positive x-direction.
~B has a magnitude of 21 and makes an angle of 30 degrees with the positive x-axis. To find its components, we can use trigonometry. The x-component of ~B is given by Bx = B * cos(theta), where B is the magnitude of ~B and theta is the angle it makes with the positive x-axis. Therefore, Bx = 21 * cos(30 degrees).
Calculating Bx, we have Bx = 21 * cos(30 degrees) = 21 * 0.866 = 18.126.
Since ~B makes an angle of 30 degrees with the positive x-axis, its y-component is given by By = B * sin(theta), where B is the magnitude of ~B and theta is the angle it makes with the positive x-axis. Therefore, By = 21 * sin(30 degrees).
Calculating By, we have By = 21 * sin(30 degrees) = 21 * 0.5 = 10.5.
Now, the components of ~B are Bx = 18.126 and By = 10.5.
To find the difference between ~A and ~B, we subtract their corresponding components:
~A - ~B = (7i) - (18.126i + 10.5j)
= (7 - 18.126)i - 10.5j
= -11.126i - 10.5j
The magnitude of ~A - ~B is given by sqrt((-11.126)^2 + (-10.5)^2).
Calculating the magnitude, we have sqrt(123.764 + 110.25) = sqrt(234.014) ≈ 15.297.
Therefore, the magnitude of ~A - ~B is approximately 15.297.