Answered

Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

Use vectors to decide whether the triangle with vertices p(1,-3,-2), q(2,0,-4), and r(6,-2,-5) is right-angled.

Sagot :

The given vertices make a right angled triangle as seen below, with the help of vectors.

What are vectors?

Vectors refer to the quantities that have both magnitude and direction but not position.

Now,

Given: Three vertices of a triangle: P(1,-3,-2),Q(2,0,-4), and R(6,-2,-5).

To find: whether they make a right angled triangle.

Finding:

Now,

  • Any two of the triangle's vectors must have a dot product of zero in order to demonstrate that the triangle is a right triangle.
  • This is because the vectors (which represent the sides) are perpendicular or orthogonal if the dot product is 0, which indicates that the angle they make is right (90°).

The triangle's three sides can be represented by the following three directional vectors:

Vector (PQ) = <2, 0, -4> - < 1, -3, -2>

=> Vector (PQ) = < 1, 3, -2>

Vector (QR) = < 6, -2, -5> - < 2, 0, -4>

=> Vector (QR) = < 4, -2, -1>

Vector (PR) = < 6, -2, -5> - <1, -3, -2>

=> Vector (PR) = < 5, 1, -3>

Now, finding a pair of vectors whose dot product = 0:

  1. Vector(PQ) . Vector(PR) = < 1, 3, -2> . < 5, 1, -3> = 5 + 3 + 6 = 14 ≠ 0
  2. Vector (QR) . Vector (PR) = < 4, -2, -1> . < 5, 1, -3> = 20 + (-2) + 3 = 21 ≠ 0
  3. Vector (PQ) . Vector (QR) = < 1, 3, -2> . < 4, -2, -1> = 4 + (-6) + 2 = 0

Hence, the sides PQ and QR must be perpendicular so that the given vertices make a right-angled triangle.

To learn more about vectors, refer to the link: https://brainly.com/question/25705666

#SPJ4

We hope this was helpful. Please come back whenever you need more information or answers to your queries. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.