Answer:
Explanation:
(a) We can find the position vectors of the points L, M, and N by taking the average of the position vectors of the points they lie between.
The midpoint of AB is given by:
N = (A + B)/2 = (41 + (41 + 21))/2 = (41 + 62)/2 = 51.5i
The midpoint of AD is given by:
L = (A + D)/2 = (41 + 6k)/2 = 23i + 3k
The midpoint of BD is given by:
M = (B + D)/2 = (41 + 21 + 6k)/2 = 31i + 3.5k
Now, we can find the vector MN by subtracting the position vectors of M and N:
MN = M - N = (31i + 3.5k) - (51.5i) = -20.5i + 3.5k
The angle between the directions of MN and OB can be found using the dot product:
cos(θ) = (MN . OB) / (|MN| * |OB|)
where θ is the angle between the two vectors. We can find the dot product and the magnitudes of the vectors as follows:
MN . OB = -20.5i + 3.5k . 41i + 21j = -20.5 * 41 + 3.5 * 21 = -847.5
|MN| = √((-20.5)^2 + (3.5)^2) = 21
|OB| = √(41^2 + 21^2) = √(1764) = 42
Substituting these values into the formula for cos(θ), we get:
cos(θ) = (-847.5) / (21 * 42) = -0.5
The angle θ can be found from the cosine inverse:
θ = cos^-1(-0.5) = 120 degrees
So, the angle between the directions of MN and OB is 120 degrees.
(b) To find the value of p, we can use the intersection of the lines through P and B and the lines through C and L. Let's call the intersection point R.
The line through P and B can be represented as P + t(B - P) = R
where t is a scalar and R is the position vector of the intersection point. Similarly, the line through C and L can be represented as:
C + s(L - C) = R
where s is a scalar. Setting these two expressions equal to each other, we get:
P + t(B - P) = C + s(L - C)
Expanding and simplifying, we get:
P + t(41i + 21j + 6k) = 21i + 6s(23i + 3k)
Comparing the i, j, and k components, we get:
P + 41t = 21 + 6s * 23
t = (21 - P)/41
6s = (P - 21)/23
s = (P - 21)/138
Substituting s back into the equation for C + s(L - C), we get:
C + (P - 21)/138 * (L - C) = R
21i + 6k + (P - 21)/138 * (23i + 3k - 6k) = R
Expanding, we get:
21i + 6k + (23/138) * Pi + (3/138) * k = R
Comparing the i and k components, we get:
21 + (23/138) * P = Ri
6 + (3/138) * P = Rk
Solving for P, we get:
P = 138 * (Ri - 21)/23 = 138 * (Rk - 6)/3
So the value of p for which the line through P and B intersects the line through C and L is given by the above formula.