If vector u = 20 m �due North and vector v =10 m � at 30 degrees East of North, find u + v and u − v.
u = (20cos90, 20sin90) = (0 , 20)
v = (10cos60, 10sin60) = (5 , 5√3)
u+v = (5 , 20 + 5√3)
u - v = (-5 , 20 - 5√3)
check by making a vector diagram sketch
I don't know 😕 🤷 😐 😔
Can you show me the steps
calculations
if vector u =20m due north and vector v =10m at 30 degrees E of N,find u + v and u - v.
To find the sum of vectors `u` and `v`, we can add their components separately.
Let's start by finding the components of vector `v`:
- The x-component of `v` is `10 cos(30°) = 8.66 m`.
- The y-component of `v` is `10 sin(30°) = 5 m`.
Now let's add the components of `u` and `v`:
- The x-component of `u + v` is `0 + 8.66 = 8.66 m`.
- The y-component of `u + v` is `20 + 5 = 25 m`.
Therefore, the sum of vectors `u` and `v` is `u + v = (8.66, 25) m`.
To find the difference of vectors `u` and `v`, we can subtract their components separately.
Let's subtract the components of `v` from `u`:
- The x-component of `u - v` is `0 - 8.66 = -8.66 m`.
- The y-component of `u - v` is `20 - 5 = 15 m`.
Therefore, the difference of vectors `u` and `v` is `u - v = (-8.66, 15) m`.
u+v= 29.1m due to 20°E.
u-v=12.4m due to 23°W
The solution you have provided is not correct.
The correct solution is:
We are given vector `u` = 20 m due North and vector `v` = 10 m at 30 degrees East of North.
To find the sum of `u` and `v`, we can add their components separately.
Let's start by finding the components of vector `v`:
- The x-component of `v` is `10 cos(30°) = 8.66 m`.
- The y-component of `v` is `10 sin(30°) = 5 m`.
Now let's add the components of `u` and `v`:
- The x-component of `u + v` is `0 + 8.66 = 8.66 m`.
- The y-component of `u + v` is `20 + 5 = 25 m`.
Therefore, the sum of vectors `u` and `v` is `u + v = (8.66, 25) m` and it makes an angle of 70 degrees with the positive x-axis.
To find the difference of `u` and `v`, we can subtract their components separately.
Let's start by finding the components of vector `u`:
- The x-component of `u` is `0`.
- The y-component of `u` is `20`.
Now let's subtract the components of `v` from `u`:
- The x-component of `u - v` is `0 - 8.66 = -8.66 m`.
- The y-component of `u - v` is `20 - 5 = 15 m`.
Therefore, the difference of vectors `u` and `v` is `u - v = (-8.66, 15) m` and it makes an angle of 123 degrees with the positive x-axis.
Note: The magnitudes of `u + v` and `u - v` are √(8.66^2 + 25^2) ≈ 27.1 m and √((-8.66)^2 + 15^2) ≈ 17.1 m, respectively.
To find the sum of two vectors u and v (u + v), we need to add their corresponding components together. Similarly, to find the difference of two vectors u and v (u - v), we need to subtract their corresponding components.
Given:
u = 20 m due North
v = 10 m at 30 degrees East of North
To find the components of vector u:
- Since u is due North, its x-component (u_x) is 0 (no displacement in the East-West direction).
- For the y-component (u_y), since it is due North, it means it is in the positive y-direction. Therefore, u_y = 20 m.
To find the components of vector v:
- Since v is at 30 degrees East of North, we first need to find its x-component (v_x).
- To do this, we need to find the projection of vector v onto the x-axis. Using trigonometry, we can calculate v_x = v * cos(30 degrees).
- Substituting the value, v_x = 10 m * cos(30 degrees) = 10 m * sqrt(3)/2 = 5*sqrt(3) m.
- For the y-component (v_y), we need to find the projection of vector v onto the y-axis. Using trigonometry, we can calculate v_y = v * sin(30 degrees).
- Substituting the value, v_y = 10 m * sin(30 degrees) = 10 m * 1/2 = 5m.
Now, let's find the sum of u and v (u + v):
- To find the x-component of (u + v), we add the x-components of u and v: (u_x + v_x) = 0 + 5*sqrt(3) m.
- To find the y-component of (u + v), we add the y-components of u and v: (u_y + v_y) = 20m + 5m = 25m.
So, vector u + v = (5*sqrt(3) m, 25 m).
Now, let's find the difference of u and v (u - v):
- To find the x-component of (u - v), we subtract the x-component of v from the x-component of u: (u_x - v_x) = 0 - 5*sqrt(3) m.
- To find the y-component of (u - v), we subtract the y-component of v from the y-component of u: (u_y - v_y) = 20m - 5m = 15m.
So, vector u - v = (-5*sqrt(3) m, 15 m).
U+v=30
u_v=10