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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

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