Consider three displacement vectors, D = (5.8 m, -6.5 m), B = (-9.1 m, 2.1 m), and A = (22 m, 18.3 m). There exist scalars a and b such that aD + bB + A = 0.
What is a?
What is b?
Well, let's solve this silly riddle!
To find a and b, we'll set up a system of equations using the components of the vectors and solve it. Here's what the equations look like:
a(5.8) + b(-9.1) + 22 = 0 (for the x-components)
a(-6.5) + b(2.1) + 18.3 = 0 (for the y-components)
Now, let's do some math!
Solving the first equation, we get:
5.8a - 9.1b + 22 = 0
Solving the second equation, we get:
-6.5a + 2.1b + 18.3 = 0
If we rearrange the equations, we have:
5.8a - 9.1b = -22
-6.5a + 2.1b = -18.3
Now, let's put our clown noses on and solve this system of equations!
Solving it (by multiplying the first equation by 2.1 and the second equation by 9.1), we get:
12.18a - 19.11b = -46.2
-59.15a + 19.11b = -166.53
Adding these two equations together, we eliminate b:
-46.97a = -212.73
Dividing both sides by -46.97, we find:
a ≈ 4.52
So, a is approximately 4.52.
Now, let's substitute this value of a into one of our original equations to find b. Let's use the first equation:
5.8(4.52) - 9.1b + 22 = 0
Simplifying this, we get:
26.216 - 9.1b = -22
Subtracting 26.216 from both sides, we have:
-9.1b = -48.216
Finally, dividing both sides by -9.1, we find:
b ≈ 5.297
So, b is approximately 5.297.
Therefore, the scalars a and b that satisfy the equation aD + bB + A = 0 are approximately a = 4.52 and b = 5.297. Have fun clowning around with vectors!
To find the scalars a and b such that aD + bB + A = 0, we need to set the sum of the vectors equal to the zero vector:
aD + bB + A = 0
Now, let's substitute the given values of D, B, and A into this equation:
a(5.8 m, -6.5 m) + b(-9.1 m, 2.1 m) + (22 m, 18.3 m) = 0
Expanding the equation, we get:
(5.8a - 9.1b + 22, -6.5a + 2.1b + 18.3) = (0, 0)
This gives us two separate equations:
5.8a - 9.1b + 22 = 0 (Equation 1)
-6.5a + 2.1b + 18.3 = 0 (Equation 2)
We can now solve these two equations to find the values of a and b. Let's start by solving Equation 2 for a:
-6.5a + 2.1b + 18.3 = 0
Rearranging the equation, we have:
-6.5a = -2.1b - 18.3
Dividing both sides by -6.5, we get:
a = (-2.1b - 18.3) / -6.5
Next, substitute this value of a into Equation 1:
5.8a - 9.1b + 22 = 0
Substituting the expression for a, we have:
5.8((-2.1b - 18.3) / -6.5) - 9.1b + 22 = 0
Now, solve for b:
5.8((-2.1b - 18.3) / -6.5) - 9.1b + 22 = 0
Simplifying the expression, we get:
(-12.18b - 106.74) / -6.5 - 9.1b + 22 = 0
Multiplying through by -6.5 to eliminate the denominator, we have:
-12.18b - 106.74 - 6.5 * 9.1b + 6.5 * 22 = 0
Simplifying further, we get:
-12.18b - 106.74 - 59.15b + 143 = 0
Combining like terms, we have:
-71.33b + 36.26 = 0
Now, solve for b:
-71.33b = -36.26
Dividing both sides by -71.33, we get:
b = 0.508
Now that we have the value of b, substitute it back into either Equation 1 or Equation 2 to find the value of a. Let's substitute it into Equation 1:
5.8a - 9.1(0.508) + 22 = 0
Simplifying the equation, we have:
5.8a - 4.623 + 22 = 0
Combine like terms:
5.8a + 17.377 = 0
Subtracting 17.377 from both sides, we get:
5.8a = -17.377
Dividing both sides by 5.8, we find:
a = -2.999
Therefore, the value of a is approximately -2.999 and the value of b is approximately 0.508.
To find the values of a and b, we need to solve the equation aD + bB + A = 0.
Let's substitute the given values for D, B, and A into the equation:
aD + bB + A = (a * 5.8 m, a * -6.5 m) + (b * -9.1 m, b * 2.1 m) + (22 m, 18.3 m) = (0, 0)
Now, let's break down the equation into its components:
a * 5.8 m + b * -9.1 m + 22 m = 0 (for the x-component)
a * -6.5 m + b * 2.1 m + 18.3 m = 0 (for the y-component)
Now we have a system of two linear equations with two unknowns (a and b). We can solve it using various methods, such as substitution or elimination. Let's solve it by elimination:
Multiplying the first equation by 6.5 and the second equation by 5.8, we get:
6.5a * 5.8 m + 6.5b * -9.1 m + 6.5 * 22 m = 0
5.8a * -6.5 m + 5.8b * 2.1 m + 5.8 * 18.3 m = 0
Now, we simplify the equations:
37.7a - 59.15b + 143 = 0
-37.7a + 12.18b + 106.14 = 0
To eliminate the "a" term, we add the two equations:
(37.7a - 37.7a) + (-59.15b + 12.18b) + (143 + 106.14) = 0
-46.97b + 249.14 = 0
Next, we isolate the variable "b":
-46.97b = -249.14
b = -249.14 / -46.97
b = 5.3
Now that we have the value of "b", we can substitute it back into one of the original equations to solve for "a". Let's choose the first equation:
37.7a - 59.15b + 143 = 0
37.7a - 59.15 * 5.3 + 143 = 0
37.7a - 312.995 + 143 = 0
37.7a - 169.995 = 0
37.7a = 169.995
a = 169.995 / 37.7
a = 4.51
Therefore, the value of "a" is 4.51 and the value of "b" is 5.3.
a( (5.8 m, -6.5 m)+b( (-9.1 m, 2.1 m)+(22 m, 18.3 m)=0
this implies
5.8a-9.1b =-22 and
-6.5a+2.1b=-18.3
solve then by any number of methods you know.