Question

whose vertices are p 2. PQRS is a parallelogram (x,y), Q (5,7), R (4,3) and S (1,2). i Express in component form PQ and SR and hence find the values of x and y. ii Calculate the magnitude of PR. li the unit vector along PR.€‹

Answer 1

The component form of vectors PQ and SR are (3, 7 - x) and (3, 1) respectively. The values of x and y are 6 and 6 respectively. The magnitude of PR is √13 and the unit vector along PR is (2/√13, -3/√13).

Step-by-step explanation:

The given parallelogram has vertices P(2, x), Q(5, 7), R(4, 3), and S(1, 2).

i) To find the component form of the vectors PQ and SR, we subtract the x-coordinates and the y-coordinates separately. So, PQ = Q - P = (5 - 2, 7 - x) = (3, 7 - x) and SR = R - S = (4 - 1, 3 - 2) = (3, 1).

Next, we can equate the x-components and y-components of the vectors to find the values of x and y. For PQ, we have 3 = 3 and 7 - x = 1. Solving these equations, we get x = 6 and y = 6.

ii) The magnitude of PR can be found using the distance formula. PR = √((4 - 2)² + (3 - 6)²) = √(4 + 9) = √13.

The unit vector along PR can be found by dividing the vector PR by its magnitude. So, the unit vector along PR is PR/|PR| = (2/√13, -3/√13).

Answer 2

P's coordinates are (2,6). The magnitude of PR is √13, and the unit vector along PR is (2/√13, -3/√13).

Step-by-step explanation:

To solve the student's question about the parallelogram PQRS, we first find the vector components of PQ and SR. Since PQRS is a parallelogram, PQ and SR are parallel and have equal magnitudes.

Given the coordinates
P (x, y),
Q (5,7),
R (4,3),
S (1,2),

Let's calculate the components of PQ and SR:

• PQ = Q - P = (5, 7) - (x, y) = (5 - x, 7 - y)
• SR = R - S = (4, 3) - (1, 2) = (3,1)

For these vectors to be equal, the components must be equal:

• 5 - x = 3 -> x = 2
• 7 - y = 1 -> y = 6

So, the coordinates for P are (2, 6).

Next, we calculate the magnitude of vector PR using the Pythagorean theorem:

• PR = R - P = (4, 3) - (2, 6) = (2, -3)
• |PR| = √(2^2 + (-3)^2) = √(4 + 9) = √13

The unit vector along PR is the vector PR divided by its magnitude:

• unit vector = PR / |PR| = (2, -3) / √13

To find the unit vector in its simplest form, we commonly write it as:

• unit vector = (2/√13, -3/√13)