A *component* of a vector is a scalar value which represents the magnitude of a vector along a certain direction. So far when we have referred to a vector's magnitude, we have been finding the magnitude along the vector's direction. If you have had previous experience with vectors, you may be familiar with finding the - and -components as shown in Figure 2-8 which represent the magnitude of the vector in the direction of the and axes respectively.

Figure 2-8.

and Components of a Vector

These values are easy to find using basic trigonometry since the axes form a right triangle: If is the angle between the x-axis and the vector and we know that the length of the hypotenuse is the magnitude of the vector , then the x-component is the length of the adjacent side and the and the y-component is the length of the opposite side: If you do not already know these relationships, you should memorize them now (keep in mind they are easy to remember if you know the definition of sine and cosine).

In linear algebra we sometimes need to find the component of a vector in a direction other than the x and y axes. In Figure 2-9, we have two vectors and and we want to know (the component of in the direction of ). Again we can form a right triangle with the two vectors and we find the following where is the angle between the two vectors: Note that this formula is the same as the formula for because in the previous example was the angle between and the x axis and in this case is the angle between and .

In linear algebra we sometimes need to find the component of a vector in a direction other than the x and y axes. In Figure 2-9, we have two vectors and and we want to know (the component of in the direction of ). Again we can form a right triangle with the two vectors and we find the following where is the angle between the two vectors: Note that this formula is the same as the formula for because in the previous example was the angle between and the x axis and in this case is the angle between and .

Figure 2-9.

Component of Vector

In linear algebra it is more common to define the component formula using the dot product. In order to do this we have to normalize to the unit vector so that the magnitude in the dot product equals 1: Substituting the geometric interpretation of the dot product into this formula gives the following which is what we obtained previously:

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