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