Problem
Let the position vectors of the points P, Q, R, and S be a = i + 2j – 5k, b = 3i + 6j + 3k, c = (17/5)i + (16/5)j + 7k, and d = 2i + j + k respectively. Then which of the following statements is true?
A The points P, Q, R, and S are NOT coplanar
B (b + 2d)/3 is the position vector of a point which divides PR internally in the ratio 5 : 4
C (b + 2d)/3 is the position vector of a point which divides PR externally in the ratio 5 : 4
D The square of the magnitude of the vector b x d is 95
Solution
We’ll pick each option one by one, and check if its true.
Option A
To check if the points are coplanar, we’ll first find any three vectors formed from the four points. I’ll take PQ, QR, and PS.
We’ll get PQ = 2i + 4j + 8k, QR = (2/5)i – (14/5)j + 4k, and PS = i – j + 6k
Now, there are two ways to proceed. One, we’ll try expressing PQ as a linear combination of QR and PS. That is, find two non-zero scalars λ and μ, such that PQ = λQR + μPS.
Or, find the scalar triple product [PQ QR PS]. If it comes out to be zero, then the four points are coplanar. Otherwise, they’re not.
I’ll use the second method. Finding the triple product is equivalent to finding this determinant: \( \begin{vmatrix} 2 & 4 & 8 \\ 2/5 & -14/5 & 4 \\ 1 & -1 & 6 \\ \end{vmatrix} \)
This comes out to be zero, implying that the points are coplanar, and option A is false. Effort wasted.
Option B
Let’s find the point that divides PR in the given ratio. We’ll use the section formula.
The required point is (5c + 4a)/9, or (21i + 24j + 15k)/9, which simplifies to (7i + 8j + 5k)/3.
And, (b + 2d)/3 equals (7i + 8j + 5k)/3 as well. So, option B is correct.
We’re kind of done here. Still, let’s evaluate the other options, for the sake of completeness.
Option C
We’ll use the section formula again. This time, the required point is 5c – 4a, or 13i + 8j + 55k, which is not equal to (b + 2d)/3, making this option incorrect.
Option D
The vector b x d is given by \( \begin{vmatrix} i & j & k \\ 3 & 6 & 3 \\ 2 & 1 & 1 \\ \end{vmatrix} \), which equals 3i + 3j – 9k.
The square of its magnitude equals 3² + 3² + (-9)² or 99, making option D incorrect as well.
Comments
A poorly designed problem. It should’ve been posed as a multiple option correct problem instead, given the effort needed for each option. For a single option correct problem, the focus should’ve been on a single concept. For example, finding the value of a parameter if the points are coplanar, and all options are different values of this parameter. Or, finding the ratio (from the options) in which a given point divides PR, and so on. Plus, quite a bit of luck involved – someone could’ve started with option B, and someone else could’ve started with A, followed by D etc. Bah, bad problem.
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