Explain why three directional vectors of the plane cannot me mutually perpendicular .
To understand why three directional vectors in a plane cannot be mutually perpendicular, we need to recall the properties of perpendicular vectors and examine the geometry of a plane.
In a two-dimensional plane, a vector represents a directional quantity with both magnitude and direction. To achieve mutual perpendicularity, three vectors must be orthogonal to each other, meaning they form right angles at their intersection points.
However, in a plane, there are only two dimensions, so it is impossible to find three vectors that are all mutually perpendicular. The reason for this lies in the fact that in two dimensions, there is only one degree of freedom for rotation. This means that once two vectors are fixed in a specific direction, the third vector will automatically lie within the span of the first two vectors.
To visualize this, consider a piece of paper representing the plane. You can set two vectors on the paper, and no matter how you rotate the paper, the third vector will always fall within the plane's surface. It will never be perpendicular to the other two vectors.
In conclusion, in a two-dimensional plane, it is not possible for three directional vectors to be mutually perpendicular due to the inherent nature and constraints of a two-dimensional space.