Answer:
Therefore, the value of μ is 6.
Explanation:
a)
i) To find the value of constants a and b, we'll equate the components of p + 2q to the given expression (a + b)i - 20j.
From p - 11i - 24j, we can determine that the i-component of p is -11 and the j-component is -24.
From q = 2i + aj, we can determine that the i-component of q is 2 and the j-component is a.
Now, let's calculate the components of p + 2q:
(i-component) = (-11) + 2(2) = -11 + 4 = -7
(j-component) = (-24) + 2(a) = -24 + 2a
We can set these components equal to the given expression:
-7 = a + b
-24 + 2a = -20
Solving the second equation, we have:
2a = -20 + 24
2a = 4
a = 2
Substituting the value of a in the first equation, we get:
-7 = 2 + b
b = -9
Therefore, the values of a and b are a = 2 and b = -9.
ii) Using the values of a = 2 and b = -9, we can find the unit vector in the direction of p + 2q.
The magnitude of p + 2q is:
√[(-7)^2 + (-24 + 2(2))^2] = √[49 + (-24 + 4)^2] = √[49 + (-20)^2] = √[49 + 400] = √449
The unit vector in the direction of p + 2q is:
[(a + b)i - 20j] / √449
= [(2 - 9)i - 20j] / √449
= (-7i - 20j) / √449
b)
The position vector of C is given by:
OC = OA + AC
Since AB:AC is 1:A, we can express AC as (1/A)AB.
Therefore, OC = OA + (1/A)AB.
c)
Given that the vector 2s + ut is parallel to the vector (+3)s + 9t, we can set up an equation of proportionality:
2s + ut = k((+3)s + 9t)
where k is the positive constant.
Expanding the equation, we get:
2s + ut = 3ks + 9kt
Comparing the coefficients of s and t, we have:
2 = 3k
u = 9k
From the first equation, we can solve for k:
3k = 2
k = 2/3
Substituting the value of k in the second equation, we get:
u = 9(2/3)
u = 6
Therefore, the value of μ is 6.