The document discusses vector operations including the dot product, cross product, and calculating the angle between vectors. It provides examples of calculating the dot product of vectors in two and three dimensions. It also gives examples of finding the angle between vectors in two and three dimensions using the dot product and vector magnitudes. The document then discusses orthogonal vectors and defines the cross product using determinants. It provides an example of calculating the cross product of two vectors and lists several properties of the cross product. Finally, it describes the right hand rule for determining the direction of the cross product.

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PavithranPuthiyapurayil ,GovernmentengineeringCollege,Kannur, Kerala

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PavithranPuthiyapurayil ,GovernmentengineeringCollege,Kannur, Kerala
Dot product calculation
The dot or scalar product of vectors and can be written as:
Example (calculation in two dimensions):
Vectors A and B are given by and . Find the dot product of the two
vectors.
Solution:
Example (calculation in three dimensions):
Vectors A and B are given by and . Find the dot product of
the two vectors.
Solution:
Calculating the Length of a Vector
The length of a vector is:
Example:
Vector A is given by . Find |A|.
Solution:
The angle between two vectors
The angle between two nonzero vectors A and B is
Example: (angle between vectors in two dimensions):
Determine the angle between and .
Solution:
We will need the magnitudes of each vector as well as the dot product.

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PavithranPuthiyapurayil ,GovernmentengineeringCollege,Kannur, Kerala
The angle is,
Example: (angle between vectors in three dimensions):
Determine the angle between and .
Solution:
Again, we need the magnitudes as well as the dot product.
The angle is,
Orthogonal vectors
If two vectors are orthogonal then: .
Example:
Determine if the following vectors are orthogonal:
Solution:
The dot product is
So, the two vectors are orthogonal.
Cross Product

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PavithranPuthiyapurayil ,GovernmentengineeringCollege,Kannur, Kerala

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PavithranPuthiyapurayil ,GovernmentengineeringCollege,Kannur, Kerala
Besides the usual addition of vectors and multiplication of vectors by scalars, there are also two types
of multiplication of vectors by other vectors. One type, the dot product, is a scalar product; the result
of the dot product of two vectors is a scalar. The other type, called the cross product, is a vector
product since it yields another vector rather than a scalar. As with the dot product, the cross product of
two vectors contains valuable information about the two vectors themselves.
The cross product of two vectors and is given by

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PavithranPuthiyapurayil ,GovernmentengineeringCollege,Kannur, Kerala
Although this may seem like a strange definition, its useful properties will soon become evident.
There is an easy way to remember the formula for the cross product by using the properties of
determinants. Recall that the determinant of a 2x2 matrix is
and the determinant of a 3x3 matrix is
Notice that we may now write the formula for the cross product as
Example 1:
The cross product of the vectors and .
Solution:
Properties of the Cross Product:
1. The length of the cross product of two vectors is
2. Anticommutativity:
3. Multiplication by scalars:

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PavithranPuthiyapurayil ,GovernmentengineeringCollege,Kannur, Kerala
4. Distributivity:
5. The scalar triple product of the vectors a, b, and c:
Which Direction?
The cross product could point in the completely opposite direction and still be at right angles to the two
other vectors, so we have the:
"Right Hand Rule"
With your right-hand, point your index finger along vector a, and point your middle finger along vector b:
the cross product goes in the direction of your thumb.

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PavithranPuthiyapurayil ,GovernmentengineeringCollege,Kannur, Kerala