Final answer:
Vertices A, B, C, and D of a parallelogram define vectors AB and CD, which are equal and opposite. Solving for m and n, we get m=6 and n=8. The magnitude of AB is √10 and the unit vector along AB (not BB, possibly a typo) is (1/√10) * (3, 1).
Step-by-step explanation:
To address the student's question about the parallelogram ABCD with given vertices, we can apply vector principles. First, we need to express vectors AB and CD in component form.
Vector AB can be found using the coordinates of A(3, 7) and B(m, n) which gives us AB = (m - 3, n - 7). Similarly, vector CD is found using C(5, 4) and D(2, 3), so CD = (2 - 5, 3 - 4) = (-3, -1).
Since ABCD is a parallelogram, AB and CD are equal and opposite vectors, meaning AB = -CD. Hence, m - 3 = 3 and n - 7 = 1, giving us m = 6 and n = 8. Now, we calculate the magnitude of AB, which is √((m - 3)² + (n - 7)²), and substituting the values of m and n, we get the magnitude of AB to be √(3² + 1²) = √10 units.
The unit vector along BB is not well defined as there is no vector BB in the parallelogram ABCD; perhaps the student meant AB. A unit vector in the direction of AB is found by dividing AB by its magnitude, resulting in (1/√10) * (3, 1).