Question

Use vectors to decide whether the triangle with vertices P(1, -5, -4), Q(2, -2, -6) , and R(6, -4, -7) is right-angled.

a.Yes, it is right-angled.
b.No, it is not right-angled.

Answer 1

By calculating the vectors for the triangle's sides and finding that the dot product of vectors PQ and QR is zero, we have confirmed that the triangle with vertices P, Q, and R is right-angled.

Step-by-step explanation:

To determine whether the triangle with vertices P(1, -5, -4), Q(2, -2, -6), and R(6, -4, -7) is right-angled, we need to calculate the vectors representing the triangle's sides and then check for a right angle between any two sides using the dot product. First, let's find the vectors PQ, PR, and QR:

• PQ = Q - P = (2-1, -2+5, -6+4) = (1, 3, -2)
• PR = R - P = (6-1, -4+5, -7+4) = (5, 1, -3)
• QR = R - Q = (6-2, -4+2, -7+6) = (4, -2, -1)

Next, we calculate the dot products of each pair of vectors.

• PQ · PR = (1)(5) + (3)(1) + (-2)(-3) = 5 + 3 + 6 = 14
• PQ · QR = (1)(4) + (3)(-2) + (-2)(-1) = 4 - 6 + 2 = 0
• PR · QR = (5)(4) + (1)(-2) + (-3)(-1) = 20 - 2 + 3 = 21

Since PQ · QR = 0, the vectors PQ and QR are perpendicular to each other, indicating that the angle between them is 90 degrees, thereby forming a right angle. Hence, the triangle is right-angled.