2 people move in the same direction along a line of equation:
(x + 3)/10 = (y + 10)/20 = (z - 10)/-20
Mobile M1 is at a point A(-3, -10, 10) and moves at a velocity of 6m/s towards point B(7, 10, -10) where Mobile 2 is found, walking in the same direction at a velocity of 3m/s. Suppose that the line equation and the coordinates are expressed in meters:
a) For each mobile, give a vector equation to determine the position at each instant t, where t is in seconds.
To find the vector equation of M1, we need to find the direction along which it is moving, which is given by the difference of coordinates of B and A:
BA=<7-(-3),10-(-10),-10-10>
=<10,20,-20>
We know that the speed (magnitude) is 3 m/s, so we normalize the direction vector and multiply by 3.
The unit vector is obtained by dividing the vector by its magnitude:
|BA|=sqrt(10^2+20^2+20^2)=30
Therefore unit vector of BA,
ba=<10/30,20/30,-20/30>=<1/3, 2/3, -2/3>
For a speed of 3 m/s, we multiply the unit vector by 3:
M1=3ba=<1,2,-2>
Since M1 start from (-3,-10,10) at time t=0, we have the vector equation as
x=<-3+t, -10+2t, 10-2t>
M2 can be calculated similarly.
THanks Mate!
You're welcome!
To determine the vector equation for the position of each mobile at any given time, we will use the formula:
P(t) = P₀ + vt
Where:
- P(t) represents the position vector at time t,
- P₀ represents the initial position vector,
- v represents the velocity vector, and
- t represents time.
Let's first calculate the velocity vectors for each mobile:
For Mobile M1:
Initial position, P₀ = A(-3, -10, 10)
Velocity vector, v = 6
For Mobile M2:
Initial position, P₀ = B(7, 10, -10)
Velocity vector, v = 3
Now, let's construct the vector equations for each mobile:
For Mobile M1:
P₁(t) = A + v₁ * t
Substituting the values:
P₁(t) = (-3, -10, 10) + 6t
For Mobile M2:
P₂(t) = B + v₂ * t
Substituting the values:
P₂(t) = (7, 10, -10) + 3t
So, the vector equation for the position of Mobile M1 at time t is P₁(t) = (-3 + 6t, -10 + 6t, 10 + 6t), and the vector equation for the position of Mobile M2 at time t is P₂(t) = (7 + 3t, 10 + 3t, -10 + 3t).