A triangle has vertices P (2,-1,0), Q (4,1,1), R (4,-5,4). Determine if the triangle is a right triangle (a) Using the Pythagorean theorem; (b) Using the properties of the dot product.

see answers below

Step-by-step explanation:

a) using the Pythagorean theorem

PQ = (2,-1,0) - (4,1,1) = (-2,-2,-1)

QR = (4,1,1) - (4,-5,4) = (0,6,-3)

RP = (4,-5,4) - (2,-1,0) = (2,-4,4)

then

length 1 = |PQ| = √[ (-2) ² + (-2) ² + (-1) ²] = √9 = 3

length 2 = |QR| = √[ (0) ² + (6) ² + (-3) ²] = √45

length 3 = |RP| = √[ (2) ² + (-4) ² + (-4) ²] = √36 = 6

then if the triangle is a right triangle

length 2 = √[ (length 1) ² + (length 3) ²] = √ (3²+6²) = √45

then our assumption is correct

b) using the properties of dot product

PQ * RP = (-2,-2,-1) * (2,-4,4) = 2 * (-2) + (-2) * (-4) + (-1) * 4 = - 4 + 8 - 4 = 0

thus PQ is perpendicular to RP. Then the triangle PQR is a right one