5. (a) Show that the points A(5, –3), B(4, –1) and C(8, –4) are collinear - Scottish Highers Maths - Question 5 - 2019

To show that points A, B, and C are collinear, we can use vector notation.

1. Find the vectors:

   • First, we will find the vector AB:

     $AB = B - A = (4 - 5, -1 - (-3)) = (-1, 2)$

   • Next, we will find the vector BC:

     $BC = C - B = (8 - 4, -4 - (-1)) = (4, -3)$

2. Check for parallelism:

   • For points to be collinear, the vectors must be scalar multiples of each other. We need to check if there exists a scalar $k$ such that:

     $AB = k imes BC$

   • This translates to the equations:

     $(-1, 2) = k imes (4, -3)$

   • This gives us two equations:

     $-1 = 4k$ $2 = -3k$

3. Solving for k:

   • From the first equation:

     $k = -\frac{1}{4}$

   • From the second equation:

     $k = -\frac{2}{3}$

   Since we get two different values for k, the vectors are not multiples of each other. However, when cross-multiplying the vector values, we find:

   $-1 \times -3 = 2 \times 4 \implies 3 = 8$ which shows they are not collinear as they do not satisfy the equations.

   Therefore, the points A, B, and C are not collinear.