If A=6i-8j,B=-8i+3j and C=26i+19j,find A and B,aA+bB+C=0
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aA+bB+C = a(6i-8j) + b(-8i+3j) + 26i+19j
So, that means you need to solve
6a-8b+26 = 0
-8a+3b+19 = 0
Now finish it off
Re_write the unit vectors in terms of column vectors, i.e,
6i-8j= (6 -8)
-8i+3j= (-8 3)
26i+19j= (26 19)
aA+bB+C=0
a(6 -8)+b(-8 3) +(26 19)=0
(6a -8a)+ (-8b 3b) +(26 19)=0
6a-8b+26=0
-8a+3b+19=0
6a-8b=-26....eqn(i)
-8a+3b=-19 ....eqn(ii)
solve using elimination method by multiplying eqn(i) by 4 and eqn(ii) by 3
24a-32b=-104
-24a+9b=-57
-23b=-161
b=7
substitute b=7 in eqn(i)
6a-8(7)=-26
6a-56=-26
6a=-26+56
6a=30
a=5
therefore, a=5 and b=7
To find the values of A and B, we can analyze the given equations and equate the corresponding coefficients.
Given:
A = 6i - 8j (Equation 1)
B = -8i + 3j (Equation 2)
Now, let's calculate aA + bB + C = 0:
aA = a(6i - 8j) = 6ai - 8aj
bB = b(-8i + 3j) = -8bi + 3bj
Combining these two terms with C:
aA + bB + C = (6ai - 8aj) + (-8bi + 3bj) + (26i + 19j)
= [6a - 8b + 26]i + [-8a + 3b + 19]j
According to the equation aA + bB + C = 0, we can equate the coefficients of i and j to zero:
6a - 8b + 26 = 0 (Equation 3)
-8a + 3b + 19 = 0 (Equation 4)
Solving equations 3 and 4 simultaneously will give us the values of a and b:
Step 1: Multiply equation 3 by 4 and equation 4 by 3 to eliminate variable 'a'.
(24a - 32b + 104) = (32a - 12b + 76)
Step 2: Simplify the equation: 8a - 20b = -28 (Equation 5)
Step 3: Subtract equation 5 from equation 4 to find 'b'.
-8a + 3b + 19 = 0
- (8a - 20b = -28)
-----------------------
23b + 47 = 28
Step 4: Solve for 'b': 23b = -19
b = -19/23
Step 5: Substitute the value of 'b' into equation 5 to find 'a'.
8a - 20(-19/23) = -28
8a + (380/23) = -28
8a = -644/23
a = -83/46
Therefore, the values of a and b are:
a = -83/46
b = -19/23
To find A and B, substitute the values of a and b into equations 1 and 2:
A = 6i - 8j
= (6)(i) - (8)(j)
= (6)(i) - (8)(j)
= (-83/46)(6)(i) - (19/23)(8)(j)
= (6)(-83/46)i + (8)(-19/23)j
= (-498/46)i + (-152/23)j
= (-249/23)i + (-76/23)j
B = -8i + 3j
= (-8)(i) + (3)(j)
= (-8)(-83/46)(i) + (3)(-19/23)(j)
= (332/46)i - (57/23)j
= (166/23)i - (57/23)j
Therefore, the values of A and B are:
A = (-249/23)i + (-76/23)j
B = (166/23)i - (57/23)j