Proof that vectors a = -4i - j, b = i - 4j, and c = 3i + 5j form a right-angled triangle:

Introduction:
To prove that vectors a, b, and c form a right-angled triangle, we need to show that the dot product of any two vectors is zero. If the dot product is zero, it implies that the vectors are perpendicular to each other, which is a property of a right-angled triangle.

Dot Product:
The dot product of two vectors a and b is given by the formula:
a · b = (a1 * b1) + (a2 * b2)

Calculating the Dot Products:
1. Dot product of vectors a and b:
a · b = (-4 * 1) + (-1 * -4)
= -4 - 4
= -8

2. Dot product of vectors b and c:
b · c = (1 * 3) + (-4 * 5)
= 3 - 20
= -17

3. Dot product of vectors c and a:
c · a = (3 * -4) + (5 * -1)
= -12 - 5
= -17

Conclusion:
Since the dot product of vectors a and b is -8, the dot product of vectors b and c is -17, and the dot product of vectors c and a is -17, we can conclude that the dot product of any two vectors is not zero. Therefore, vectors a, b, and c do not form a right-angled triangle.

Counterexample:
To further illustrate the counterexample, we can calculate the magnitudes of the vectors:
1. Magnitude of vector a:
|a| = √((-4)^2 + (-1)^2)
= √(16 + 1)
= √17

2. Magnitude of vector b:
|b| = √(1^2 + (-4)^2)
= √(1 + 16)
= √17

3. Magnitude of vector c:
|c| = √(3^2 + 5^2)
= √(9 + 25)
= √34

If vectors a, b, and c formed a right-angled triangle, the Pythagorean theorem would hold, and the sum of the squares of the magnitudes of two smaller vectors would be equal to the square of the magnitude of the largest vector. However, in this case, |a|^2 + |b|^2 ≠ |c|^2, which further confirms that vectors a, b, and c do not form a right-angled triangle.

Therefore, vectors a = -4i - j, b = i - 4j, and c = 3i + 5j do not form a right-angled triangle.