# What is the vector that represents a point at a distance of 5 units and lying on the x-y plane, if the line drawn from the point to the origin makes an angle of 45 degrees with the x-axis.

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I guess you would like to know the vector between the origin and a point A. A lies at a distance of 5 units from the origin, is located on the x-y plane and the line joining A and the origin makes an angle of 45 degrees with the x-axis. It is not specified whether you are referring to the positive x-axis or the negative.

Taking all combinations, there are 4 possible vectors.

The magnitude of all the possible vectors is 5 units.

If the line makes an angle of 45 degrees with reference to the positive x-axis, it can make an angle of 45 degrees with the positive y-axis or an angle of 135 degrees with the positive y-axis. This gives the vectors: sqrt(5/2))*i + sqrt(5/2)*j and sqrt(5/2)i - sqrt(5/2)*j

If the angle of 45 degrees is with reference to the negative x-axis, the vectors can be -sqrt(5/2)*i + sqrt(5/2)*j and -sqrt(5/2) - sqrt(5/2)*j

**There are 4 possible vectors that satisfy the given criteria: sqrt(5/2))*i + sqrt(5/2)*j , sqrt(5/2)i - sqrt(5/2)*j , -sqrt(5/2)*i + sqrt(5/2)*j and -sqrt(5/2) - sqrt(5/2)*j**