Determining the Collinearity of Vectors in a Mathematical Context

In the realm of vector mathematics, the concept of collinearity plays a crucial role. Collinearity refers to the property of two vectors pointing in the same or exactly opposite directions. This article will delve into a specific problem where we need to determine the value of x for which the vectors c (x - 2)ab and d 2x - 1a - b are collinear. The solution involves a series of logical steps that will be explained in detail, making it a valuable resource for students and professionals alike in the field of vector analysis.

Understanding the Problem

The vectors c and d are defined as follows:

c (x - 2)ab d 2x - 1a - b

Two vectors c and d are collinear if there exists a scalar k such that c k * d. In this case, we need to find the value of x that satisfies this condition.

Step-by-Step Solution Process

Step 1: Set Up the Equation

Given the definitions of c and d, we can set up the equation:

x - 2ab k2x - 1a - b

Step 2: Expand Both Sides

Expanding both sides gives:

x - 2a × x - 2b k2x - 1a - k2x - 1b

Step 3: Rearrange the Equation

Now we can rearrange the equation to group terms involving a and b:

[x - 2 - k2x - 1]a × [x - 2 k2x - 1]b 0

Step 4: Set Coefficients to Zero

For the equation to hold true for non-collinear vectors a and b, the coefficients of a and b must both equal zero:

Coefficient of a: x - 2 - k2x - 1 0 Coefficient of b: x - 2 k2x - 1 0

Step 5: Solve the System of Equations

From the first equation:

k2x - 1 x - 2

From the second equation:

k2x - 1 2 - x

Since both expressions equal k2x - 1, we can set them equal to each other:

x - 2 2 - x

Step 6: Solve for x

Now solve for x:

x - 2 - x 2

2x - 2 2

2x 4

x 2

Step 7: Verify

To verify, substitute x 2 back into either equation for k:

From the first equation: k22 - 1 2 - 2 k5 0 k 0

From the second equation:

k22 - 1 2 - 2 k5 0 k 0

Both conditions are satisfied.

Conclusion

Thus, the value of x for which the vectors c and d are collinear is:

x 2

It's worth noting that ab and a - b cannot be parallel as long as a and b are non-zero and non-collinear vectors. Hence, c and d can only be collinear if they are zero vectors, which is a special case. The only possibility is:

x 2 or x -1/2

Further Considerations

Considering the vectors c and d are collinear, one vector must be a scalar multiple of the other. Therefore, let:

d k * c

Where k is a scalar. This results in:

2x - 1a - b k * (x - 2)ab

Comparing the coefficients of a and b, we get:

2x - 1 / x - 2 k -1 / x - 2 k

Setting the two expressions for k equal to each other:

2x - 1x - 2 -1x - 2

Further simplification leads us to:

x 2 or x -1/2

Final Conclusion

The value of x for which the vectors c (x - 2)ab and d 2x - 1a - b are collinear is:

x 2 or x -1/2