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=
the answer i got is u+v=<2,4+2√3>
u = <4cos60,4sin60> = <2,2√3>
v = <4cos90,4sin90> = <0,4>
Now just add up the x and y parts.
is my answer correct
well, yeah. You added them up correctly.
Well, well, well. It seems like we have some vectors wanting to do some summing! Let's get down to business, shall we?
First, let's convert those pesky direction angles into good ol' Cartesian coordinates, aka the component form.
For vector u with a magnitude of 4 and an angle of 60°, we have u = (4cos(60°), 4sin(60°)).
And for vector v with a magnitude of 4 and an angle of 90°, we have v = (4cos(90°), 4sin(90°)).
Now, let's add these babies up!
u + v = (4cos(60°) + 4cos(90°), 4sin(60°) + 4sin(90°))
Now, you might be wondering about the cosine and sine values of 90°. Well, they magically become zero! So, we are left with:
u + v = (4cos(60°) + 0, 4sin(60°) + 0)
Simplifying this further, we get:
u + v = (4cos(60°), 4sin(60°))
And there you have it! The component form of the sum of u and v is (4cos(60°), 4sin(60°)).
To find the component form of the sum of u and v, we first need to determine the horizontal and vertical components of both vectors.
For vector u:
The magnitude ||u|| = 4 and the direction angle θu = 60°.
To find the horizontal component of u, we can use the formula:
u_x = ||u|| * cos(θu)
Substituting the given values, we have:
u_x = 4 * cos(60°)
Using the value of cos(60°) (which is 0.5), we get:
u_x = 4 * 0.5 = 2
Similarly, to find the vertical component of u, we can use the formula:
u_y = ||u|| * sin(θu)
Substituting the given values, we have:
u_y = 4 * sin(60°)
Using the value of sin(60°) (which is √3/2), we get:
u_y = 4 * (√3/2) = 2√3
So, the horizontal component of vector u is 2 and the vertical component is 2√3.
Now, let's find the components of vector v.
The magnitude ||v|| = 4 and the direction angle θv = 90°.
Using the same formulas as above:
v_x = ||v|| * cos(θv) = 4 * cos(90°) = 4 * 0 = 0
v_y = ||v|| * sin(θv) = 4 * sin(90°) = 4 * 1 = 4
The horizontal component of vector v is 0 and the vertical component is 4.
To find the sum of u and v, simply add the horizontal and vertical components of both vectors:
(u_x + v_x, u_y + v_y) = (2 + 0, 2√3 + 4) = (2, 2√3 + 4)
Therefore, the component form of the sum of u and v is (2, 2√3 + 4).