If D = (3.00î − 4.00ĵ) m, B = (−5.00î + 4.00ĵ) m, and A = (18.0î − 8.00ĵ) m, find the unknown constants a and b such that aD + bB + A = 0.
Well, in order to find the unknown constants a and b, let's set up the equation aD + bB + A = 0 and solve it step by step.
We start by substituting the given values into the equation:
a(3.00î − 4.00ĵ) + b(−5.00î + 4.00ĵ) + (18.0î − 8.00ĵ) = 0
Now let's group the terms with the same components together:
(3.00a - 5.00b + 18.0)î + (-4.00a + 4.00b - 8.00)ĵ = 0
Since the left side of the equation equals zero, the coefficients of the î and ĵ components must also be zero:
3.00a - 5.00b + 18.0 = 0 (1)
-4.00a + 4.00b - 8.00 = 0 (2)
Now we have a system of two linear equations. Let's solve it.
From equation (1), we can express a in terms of b:
3.00a = 5.00b - 18.0
a = (5.00b - 18.0) / 3.00
Substituting this value into equation (2):
-4.00((5.00b - 18.0) / 3.00) + 4.00b - 8.00 = 0
Now we can solve for b:
-20.00b + 72.00 + 12.00b - 24.00 = 0
-8.00b + 48.00 = 0
-8.00b = -48.00
b = (-48.00) / (-8.00)
b = 6.00
Now that we know the value of b, we can substitute it back into equation (1) to solve for a:
3.00a = 5.00(6.00) - 18.0
3.00a = 30.00 - 18.0
3.00a = 12.00
a = 12.00 / 3.00
a = 4.00
So, the unknown constants are a = 4.00 and b = 6.00.
Therefore, aD + bB + A = 4.00(3.00î − 4.00ĵ) + 6.00(−5.00î + 4.00ĵ) + (18.0î − 8.00ĵ) m = 0
To find the unknown constants a and b, we can set up a system of equations using the given vector equation:
aD + bB + A = 0
Let's compare the components on both sides of the equation:
For the x-component:
a(3.00) + b(-5.00) + 18.0 = 0
For the y-component:
a(-4.00) + b(4.00) - 8.00 = 0
Now we can solve this system of equations to find the values of a and b.
To find the unknown constants a and b, we can set up a system of equations using the given information.
Given:
D = (3.00î - 4.00ĵ) m (vector)
B = (-5.00î + 4.00ĵ) m (vector)
A = (18.0î - 8.00ĵ) m (vector)
We are looking for constants a and b such that:
aD + bB + A = 0
Let's break down the equation component-wise and set up the system of equations:
Component-wise equation:
a(3.00î - 4.00ĵ) + b(-5.00î + 4.00ĵ) + (18.0î - 8.00ĵ) = 0
Breaking down the equation for the x-component:
3.00a - 5.00b + 18.0 = 0
Breaking down the equation for the y-component:
-4.00a + 4.00b - 8.00 = 0
Now we have a system of two equations:
1) 3.00a - 5.00b + 18.0 = 0
2) -4.00a + 4.00b - 8.00 = 0
Now we can solve this system of equations to find the values of a and b.
To solve this system, we can use the method of substitution or elimination. Let's use the method of elimination:
Multiply the first equation by 4 and the second equation by 3 to eliminate the 'a' term:
4(3.00a - 5.00b) + 3(-4.00a + 4.00b) = 4(0) + 3(0)
Simplifying the equation:
12.00a - 20.00b - 12.00a + 12.00b = 0
Combine like terms:
-8.00b = 0
Solving for b:
b = 0
Now substitute the value of b back into one of the original equations (let's use the first one) to solve for a:
3.00a - 5.00(0) + 18.0 = 0
3.00a + 18.0 = 0
3.00a = -18.0
a = -18.0 / 3.00
a = -6.00
Therefore, the values of a and b that satisfy the equation aD + bB + A = 0 are:
a = -6.00
b = 0
Just set up equations for the components:
3a - 5b + 18 = 0
-4a + 4b - 8 = 0
Now just solve for a and b.