Answered

Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.

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

Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.