Find the component form of the sum of u and v with the given magnitudes and direction angles θu and θv.
Magnitude Angle
||u||=4 θu = 60°
||v||=4 θu = 90°
u+v=
To find the component form of the sum of u and v, we'll need to break down each vector into its x and y components, and then add the corresponding x and y components together.
Let's start by finding the x and y components of vector u. We can use the magnitude and direction angle to calculate these components:
For vector u:
Magnitude (||u||) = 4
Direction angle (θu) = 60°
The x component (ux)can be calculated using the formula:
ux = ||u|| * cos(θu)
Substituting the values, we get:
ux = 4 * cos(60°) = 4 * 0.5 = 2
The y component (uy) can be calculated using the formula:
uy = ||u|| * sin(θu)
Substituting the values, we get:
uy = 4 * sin(60°) = 4 * √3/2 = 4 * 1.732/2 ≈ 6.928
So the x and y components of vector u are: ux = 2 and uy ≈ 6.928.
Now let's find the x and y components of vector v. Similar to vector u, we can use the magnitude and direction angle to calculate these components:
For vector v:
Magnitude (||v||) = 4
Direction angle (θv) = 90°
The x component (vx) can be calculated using the formula:
vx = ||v|| * cos(θv)
Substituting the values, we get:
vx = 4 * cos(90°) = 4 * 0 = 0
The y component (vy) can be calculated using the formula:
vy = ||v|| * sin(θv)
Substituting the values, we get:
vy = 4 * sin(90°) = 4 * 1 = 4
So the x and y components of vector v are: vx = 0 and vy = 4.
Finally, to find the sum (u+v), we add the corresponding x and y components together:
(u+v) = (ux + vx)i + (uy + vy)j = (2 + 0)i + (6.928 + 4)j = 2i + 10.928j
Therefore, the component form of the sum of u and v is 2i + 10.928j.