PROVE THAT SCALAR PRODUCT OF 2 PERPENDICULAR VECTORS IS ZERO
Asked by rksvikhyat
| 1st Jun, 2012,
09:39: AM
In Euclidean geometry, the dot product of vectors expressed in an orthonormal basis is related to their length and angle. For such a vector
, the dot product
is the square of the length (magnitude) of
, denoted by
:

If
is another such vector, and
is the angle between them:

This formula can be rearranged to determine the size of the angle between two nonzero vectors:

The CauchySchwarz inequality guarantees that the argument of
is valid.
One can also first convert the vectors to unit vectors by dividing by their magnitude:

then the angle
is given by

As the cosine of 90° is zero, the dot product of two orthogonal vectors is always zero. Moreover, two vectors can be considered orthogonal if and only if their dot product is zero, and they have non-null length. This property provides a simple method to test the condition of orthogonality.

In Euclidean geometry, the dot product of vectors expressed in an orthonormal basis is related to their length and angle. For such a vector , the dot product
is the square of the length (magnitude) of
, denoted by
:
If is another such vector, and
is the angle between them:
This formula can be rearranged to determine the size of the angle between two nonzero vectors:
The CauchySchwarz inequality guarantees that the argument of is valid.
One can also first convert the vectors to unit vectors by dividing by their magnitude:
then the angle is given by
Answered by
| 1st Jun, 2012,
09:49: AM
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