Question

Determine the location of a point \((x, y, z)\) that satisfies the condition(s). \(x y>0, \quad z=-3\)

Short answer

The point \((x, y, z)\) satisfying the given conditions is located either in the first quadrant (where \(x\) and \(y\) are both positive) or in the third quadrant (where \(x\) and \(y\) are both negative), with the \(z\)-coordinate being -3 in both cases.

Step-by-step solution

Understand Conditions

The conditions given in the problem are \(x y>0\) and \(z=-3\). The first condition implies that the product of \(x\) and \(y\) should be positive, which is only possible when \(x\) and \(y\) have the same sign; i.e., they are either both positive or both negative. The second condition directly states that \(z\) is -3.

Apply First Condition

When \(x y>0\), this means that \(x\) and \(y\) are either both positive or both negative. So, the point \((x, y, z)\) is in either the first quadrant (where \(x\) and \(y\) are both positive) or the third quadrant (where \(x\) and \(y\) are both negative) in the xy-plane.

Apply Second Condition

The \(z=-3\) condition is straightforward. Regardless of the position of the point \((x, y)\) in the xy-plane, its \(z\)-coordinate is always -3.

Combine Conditions

Combining the results from step 2 and step 3, the point \((x, y, z)\) satisfying the given conditions is found either in the positive x and y region with \(z=-3\) (first quadrant), or in the negative x and y region with \(z=-3\) (third quadrant).