For what value of m are the point with position vectors 10i + 3j, 12i 5j and mi + 11j collinear?a)8b)4c)8d)12Correct answer is option 'C'. Can you explain this answer?

Question

For what value of m are the point with position vectors 10i + 3j, 12i  5j and mi + 11j collinear?a)8b)4c)8d)12Correct answer is option 'C'. Can you explain this answer?

Answer

Given:
- The position vectors of three points are given as follows:
- A: 10i + 3j
- B: 12i + 5j
- C: mi + 11j
- The points A, B, and C are collinear.

To find:
The value of m.

Explanation:
Two vectors are collinear if they are scalar multiples of each other. Therefore, the vector from A to B should be a scalar multiple of the vector from A to C.

Step 1: Finding the vector from A to B
The vector from A to B is given by subtracting the position vector of A from the position vector of B:
Vector AB = B - A = (12i + 5j) - (10i + 3j) = 2i + 2j

Step 2: Finding the vector from A to C
The vector from A to C is given by subtracting the position vector of A from the position vector of C:
Vector AC = C - A = (mi + 11j) - (10i + 3j) = (m-10)i + (11-3)j = (m-10)i + 8j

Step 3: Checking for collinearity
Since A, B, and C are collinear, the vector AB should be a scalar multiple of the vector AC. In other words, the x and y components of both vectors should be proportional.

Comparing the x components:
2 = (m-10) * k, where k is the scalar multiple

Comparing the y components:
2 = 8 * k

Step 4: Solving for m
From the y component equation, we can determine the value of k:
2 = 8 * k
k = 2/8
k = 1/4

Substituting the value of k into the x component equation:
2 = (m-10) * (1/4)
8 = m - 10
m = 8 + 10
m = 18

Conclusion:
The value of m is 18. However, none of the given options match this value. Therefore, the correct answer is not among the options provided.

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