If vector A = (6.00 i -8.00 j) units, vector B = (-8.00 i + 3.00 j) units, and vector C = (25.0 i + 22.0 j) units, determine a and b such that (a vector A + b vector B + vector C= 0.)
a=?
b=?
Well, let's solve this equation step by step, but don't worry, I won't be too "vectorious" with you!
First, we'll write the equation with the given vectors:
(a * vector A) + (b * vector B) + vector C = 0
Now, let's break it down into its components:
(a * 6.00 i - 8.00 j) + (b * -8.00 i + 3.00 j) + (25.0 i + 22.0 j) = 0
Next, let's combine similar components:
(6.00a - 8.00b + 25.0) i + (-8.00a + 3.00b + 22.0) j = 0
Since the equation needs to hold true, the coefficients of the i and j terms should be equal to 0:
6.00a - 8.00b + 25.0 = 0 (Equation 1)
-8.00a + 3.00b + 22.0 = 0 (Equation 2)
Now, we have a system of two linear equations. Time to put on my solving hat! 🎩
From Equation 1, we rearrange it to solve for a:
6.00a = 8.00b - 25.0
a = (8.00b - 25.0) / 6.00
Now, substitute this value of a into Equation 2:
-8.00((8.00b - 25.0) / 6.00) + 3.00b + 22.0 = 0
Oh boy, time for some calculations! Simplifying this equation will give us a value for b:
-64/6 * b + 200/6 - 3/2 * b + 44/2 = 0
-(32/3) * b + (100/3) - (9/2) * b + 22 = 0
-(64/6) * b - (54/6) * b = -(100/3) - 22
-(118/6) * b = -(100 + 66)/3
b = (166/6) / (118/6)
b = 83/59
Great! We have found the value of b. Now let's substitute it back into Equation 1 to find a:
6.00a = 8.00(83/59) - 25.0
6.00a = 996/59 - 25.0
6.00a = 996/59 - 1475/59
6.00a = -(479/59)
a = -(479/354)
So, to wrap it up, we have found that a ≈ -(479/354) and b ≈ 83/59.
Remember, these values might seem a bit "abstract," but we've done the math!
To determine values for a and b such that (a vector A + b vector B + vector C= 0), we need to set up a system of equations and solve for a and b.
First, let's write out the given equation:
a vector A + b vector B + vector C = 0
Now let's break it down into its x and y components:
For the x-component:
a(6.00) + b(-8.00) + 25.0 = 0
And for the y-component:
a(-8.00) + b(3.00) + 22.0 = 0
Now we have two equations:
6a - 8b + 25 = 0 (equation 1)
-8a + 3b + 22 = 0 (equation 2)
To solve this system of equations, we can use any method such as substitution or elimination. Let's use the elimination method in this case.
Multiply equation 1 by 8 and equation 2 by 6 to make the coefficients of 'a' equal:
48a - 64b + 200 = 0 (equation 3)
-48a + 18b + 132 = 0 (equation 4)
Add equation 3 and equation 4 to eliminate 'a' terms:
-46b + 332 = 0
Now, isolate 'b':
-46b = -332
b = -332 / -46
b = 7.2174
Substitute the found value of 'b' into either equation 1 or equation 2 to solve for 'a'. Let's use equation 1:
6a - 8(7.2174) + 25 = 0
6a - 57.7392 + 25 = 0
6a = 32.7392
a = 32.7392 / 6
a = 5.4565
So, the values of a and b that satisfy the equation (a vector A + b vector B + vector C = 0) are:
a = 5.4565
b = 7.2174
But it is wrong
Ffyytttt
x direction:
6 a - 8 b + 25 = 0
y direction:
-8 a + 3 b + 22 = 0
==========================
24 a - 32 b + 100 = 0
-24 a + 9 b + 66 = 0
-------------------------add
- 23 b + 166 = 0
b = 7.22
a = 5.46
check my arithmetic !