Explain why three directional vectors of the plane cannot me mutually perpendicular .
If three vector are mutually perpendicular, they form a basis for a three dimensional space. A plane is a two-dimensional space, so it cannot have three mutually perpendicular vectors.
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To understand why three directional vectors of a plane cannot be mutually perpendicular, let's first define what it means for vectors to be mutually perpendicular.
In a three-dimensional space, two vectors are said to be mutually perpendicular or orthogonal if their dot product is zero. The dot product of two vectors is calculated by multiplying their corresponding components and then summing the products.
Now, let's consider a plane in three-dimensional space. A plane is a two-dimensional surface represented by a flat, infinite sheet. It is determined by a point and two non-parallel vectors that lie within the plane.
If we have three vectors that are mutually perpendicular, it means that each vector is orthogonal to the other two. Therefore, the dot product of each pair of vectors should be zero.
However, in a plane, there are only two independent directions. This means that two non-parallel vectors can span the entire plane, and any additional vector in that plane can be expressed as a linear combination of the two original vectors.
When we have three vectors in a plane, we can always find a linear relationship between them, meaning one vector can be written as a combination of the other two. In this case, it is impossible for all three vectors to be mutually perpendicular because one vector would not be orthogonal to the other two.
In summary, three directional vectors in a plane cannot be mutually perpendicular because a plane is a two-dimensional space, and any three vectors in that plane can always be expressed as a linear combination of the other two.