Question

identify the graph for the point ( e(-1,3,-1) ) in three - dimensional space.

Explanation:

Step1: Analyze the coordinates

In three - dimensional space, for a point \(E(x,y,z)\), \(x=-1\) (negative \(x\) - direction), \(y = 3\) (positive \(y\) - direction), \(z=-1\) (negative \(z\) - direction).

Step2: Check the first graph

In the first graph, the \(x\) and \(z\) coordinates are negative and \(y\) is positive. The dashed lines represent the movement from the origin: \(x=-1\) (left along \(x\) - axis), \(y = 3\) (not in the positive \(y\) direction as per the graph's dashed line mis - representation for \(y\) in this analysis of wrong graphs), \(z=-1\) (down along \(z\) - axis).

Step3: Check the second graph

In the second graph, the \(x\) coordinate is not \(- 1\) (the movement from the origin for \(x\) is wrong as per the point \(E(-1,3,-1)\) where \(x=-1\) should be a left - ward movement from the origin, but in this graph, the \(x\) part of the dashed line is incorrect for \(x=-1\)).

Step4: Check the third graph

In the third graph, the \(z\) coordinate is not \(-1\) (the movement along \(z\) - axis is wrong as \(z = - 1\) should be a downward movement from the origin, but in this graph, the \(z\) part of the dashed line is incorrect).

Step5: Check the fourth graph

In the fourth graph:

• For the \(x\) - coordinate: Move \(1\) unit in the negative \(x\) - direction (left).
• For the \(y\) - coordinate: Move \(3\) units in the positive \(y\) - direction (right).
• For the \(z\) - coordinate: Move \(1\) unit in the negative \(z\) - direction (down). The dashed lines correctly represent the movement from the origin to the point \(E(-1,3,-1)\)

Answer:

The fourth graph.