Question

In Exercises 61 and \(62,\) use vectors to determine whether the points are collinear. (0,-2,-5),(3,4,4),(2,2,1)

Short answer

Yes, the given points are collinear.

Step-by-step solution

Define The Points

The three given points are A = (0, -2, -5), B = (3, 4, 4), C = (2, 2, 1). We will represent these points by vectors in 3-dimensional space.

Calculate Vectors

Calculate the vectors AB and BC. The vector AB = B - A = (3,4,4) - (0,-2,-5) = (3,6,9) and the vector BC = C - B = (2,2,1) - (3,4,4) = (-1,-2,-3).

Cross Product of Vectors

Now, compute the cross product of vectors AB x BC. The cross product of two vectors A = (a1, b1, c1) and B = (a2, b2, c2) is given by: AB X BC = ((6*-3 - 9*-2), -(3*-3 - 9*-1), (3*-2 - 6*-1)) = (0,0,0).

Determine Collinearity

Now that we've computed the cross product, check the result. It's evident from our computation that AB x BC is a zero vector. In such a case, all three points are said to be collinear.