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VECTOR METHODS
Areas of focus:
- Vectors and vector addition
- Unit vectors
- Base vectors and vector components
- Rectangular coordinates in 2-D
- Rectangular coordinates in 3-D
- A vector connecting two points
- Dot product
- Cross product
- Triple product
- Triple vector product
Vectors and vector addition:
A scalar is a quantity like mass or temperature that only has a magnitude. On the other hand, a vector is a mathematical object that has magnitude and direction. A line of given length and pointing along a given direction, such as an arrow, is the typical representation of a vector. Typical notation to designate a vector is a boldfaced character, a character
with and arrow on it, or a character with a line under it (i.e., ). The
magnitude of a vector is its length and is normally denoted by or A.
Addition of two vectors is accomplished by laying the vectors head to tail in sequence to create a triangle such as is shown in the figure.
The following rules apply in vector algebra.
where P and Q are vectors and a is a scalar.
Unit vectors:
A unit vector is a vector of unit length. A unit vector is sometimes denoted by replacing the arrow on a vector with a "^" or just adding a "^" on a boldfaced character (i.e.,
). Therefore,
Each one of the vectors u 1 , u 2 , and u 3 is parallel to one of the base vectors and can be written as scalar multiple of that base. Let u 1 , u 2 , and u 3 denote these scalar multipliers such that one has
The original vector u can now be written as
The scalar multipliers u 1 , u 2 , and u 3 are known as the components of u in the base described by the base vectors e 1 , e 2 , and e 3. If the base vectors are unit vectors, then the components represent the lengths, respectively, of the three vectors u 1 , u 2 , and u 3. If the base vectors are unit vectors and are mutually orthogonal, then the base is known as an orthonormal, Euclidean, or Cartesian base.
A vector can be resolved along any two directions in a plane containing it. The figure shows how the parallelogram rule is used to construct vectors a and b that add up to c.
In three dimensions, a vector can be resolved along any three non-coplanar lines. The figure shows how a vector can be resolved along the three directions by first finding a vector in the plane of two of the directions and then resolving this new vector along the two directions in the plane.
Due to the orthogonality of the bases, one has the following relations.
Rectangular coordinates in 3-D:
The base vectors of a rectangular coordinate system are given by a set of three mutually
orthogonal unit vectors denoted by , , and that are along the x , y , and z coordinate directions, respectively, as shown in the figure.
The system shown is a right-handed system since the thumb of the right hand points in the direction of z if the fingers are such that they represent a rotation around the z -axis from x to y. This system can be changed into a left-handed system by reversing the direction of any one of the coordinate lines and its associated base vector.
In a rectangular coordinate system the components of the vector are the projections of the vector along the x , y , and z directions. For example, in the figure the projections of vector A along the x, y, and z directions are given by Ax, Ay, and Az , respectively.
where the angles , , and are the angles shown in the figure. As shown in the figure, the direction cosines represent the cosines of the angles made between the vector and the three coordinate directions.
The direction cosines can be calculated from the components of the vector and its magnitude through the relations
The three direction cosines are not independent and must satisfy the relation
This results form the fact that
A unit vector can be constructed along a vector using the direction cosines as its components along the x , y , and z directions. For example, the unit- vector along the vector A is obtained from
Therefore,
A vector connecting two points:
Dot product:
The dot product is denoted by " " between two vectors. The dot product of vectors A and B results in a scalar given by the relation
where is the angle between the two vectors. Order is not important in the dot product as can be seen by the dot products definition. As a result one gets
The dot product has the following properties.
Since the cosine of 90o^ is zero, the dot product of two orthogonal vectors will result in zero.
Since the angle between a vector and itself is zero, and the cosine of zero is one, the magnitude of a vector can be written in terms of the dot product using the rule
Rectangular coordinates:
When working with vectors represented in a rectangular coordinate system by the components
then the dot product can be evaluated from the relation
This can be verified by direct multiplication of the vectors and noting that due to the orthogonality of the base vectors of a rectangular system one has
The vector projection of A along the unit vector simply multiplies the scalar projection by the unit vector to get a vector along. This gives the relation
The cross product:
The cross product of vectors a and b is a vector perpendicular to both a and b and has a magnitude equal to the area of the parallelogram generated from a and b. The direction of the cross product is given by the right-hand rule. The cross product is denoted by a " " between the vectors
Order is important in the cross product. If the order of operations changes in a cross product the direction of the resulting vector is reversed. That is,
The cross product has the following properties.
Rectangular coordinates:
When working in rectangular coordinate systems, the cross product of vectors a and b given by
can be evaluated using the rule
One can also use direct multiplication of the base vectors using the relations
The triple product has the following properties
Rectangular coordinates:
Consider vectors described in a rectangular coordinate system as
The triple product can be evaluated using the relation
Triple vector product:
The triple vector product has the properties
