Tip:
Highlight text to annotate it
X
Consider the three dimensional vector, V, somewhere in space.
The magnitude of the vector, which is its length, is identified by the letter "V" without a vector arrow above it.
To define the direction of vector V, vector notation may be used.
The combined effect of the components of vector V on a body are equivalent to the effect of the original vector.
To compute the magnitude of the x-component, vector V is projected onto the x-axis.
Notice the right-triangle in the plane formed by the two vectors.
By referencing the angle between the vectors, which is the directional angle alpha,
the x-component of vector V may be calculated using the cosine of alpha, which is the directional cosine of vector V.
Likewise, the y-component of vector V may be calculated using the cosine of beta as the directional cosine of vector V.
And the z-component of vector V may be calculated using the cosine of gamma as the directional cosine of vector V.
So, vector V may be written in terms of its directional cosines.
Another way to define the direction of vector V is to use a projection onto the x-y plane, like a shadow.
Since the z-axis is perpendicular to the x-y plane,
the angle gamma is used to compute the length of vector V projected onto the x-y plane.
Now measuring the angle theta from the x-axis in a counter-clockwise direction to the projection,
The x-componet and y-component of vector V may be calculated.
So vector V may be specified by its magnitude and angles theta and gamma, which is standard for spherical coordinates.
Given a vector in space, first establish the coordinate system for referencing vector components.
The magnitude of vector A equals the squareroot of the sum of the squares of each component.
Now compute the direction angles.
Now the magnitude of vector A and the direction angles can be substituted into the expanded vector notation.
In some applications, it may be convienent to substitute the value of the direction cosines.
To see more on this, review 3-D unit vectors.