**Introduction to Vectors in Cartesian Three-Space**

A *vector* in Cartesian three-space is the same as a vector in the Cartesian plane, except that there is more “freedom” in terms of direction. This makes the expression of direction in 3D more complicated than is the case in 2D. It also makes vector arithmetic a lot more interesting!

**Equivalent Vectors**

In Cartesian three-space, vectors **a** and **b** can be denoted as arrow-tipped line segments from the origin (0,0,0) to points ( *x* _{a} , *y* _{a} , *z* _{a} ) and ( *x* _{b} , *y* _{b} , *z* _{b} ), as shown in Fig. 9-5. This, like all three-space drawings in this chapter, is a perspective illustration. Both vectors in this example point in directions on the reader’s side of the plane containing the page. In a true 3D model, both of them would “stick up out of the paper at an angle.”

In Fig. 9-5, both vectors **a** and **b** have their end points at the origin. This is the standard form of a vector in any coordinate system. In order for the following formulas to hold, vectors must be expressed in standard form. If a given vector is not in standard form, it can be converted by subtracting the coordinates ( *x* _{0} , *y* _{0} , *z* _{0} ) of the starting point from the coordinates of the end point ( *x* _{1} , *y* _{1} , *z* _{1} ). For example, if a vector **a** * starts at (4,7,0) and ends at (1,−3,5), it reduces to an equivalent vector **a** in standard form:

**a** = [(1 − 4), (−3 − 7), (5 − 0)]

= (−3, −10, 5)

Any vector **a** *, which is parallel to **a** and has the same length as **a** , is equal to vector **a** , because a* has the same magnitude and the same direction as **a** . Similarly, any vector **b** *, which is parallel to **b** and has the same length as **b** , is defined as being equal to **b** . As in the 2D case, a vector is defined solely on the basis of its magnitude and its direction. Neither of these two properties depends on the location of the end point.

**Defining the Magnitude and Direction of a Vector**

**Defining The Magnitude**

When the end point of a vector **a** is at the origin, the magnitude of **a** , written | **a** | or *a* , can be found by a three-dimensional extension of the Pythagorean theorem for right triangles. The formula looks like this:

The magnitude of any vector **a** in standard form is simply the distance of the end point from the origin. Note that the above formula is the distance formula for two points, (0,0,0) and ( *x* _{a} , *y* _{a} , *z* _{a} ).

**Direction Angles And Cosines**

The direction of a vector **a** in standard form can be defined by specifying the angles *θ* _{x} , *θ* _{y} , and *θ* _{z} that the vector **a** subtends relative to the positive *x* , *y* , and *z* axes respectively (Fig. 9-6). These angles, expressed in radians as an ordered triple ( *θ* _{x} , *θ* _{y} , *θ* _{z} ), are the *direction angles* of **a** .

Sometimes the cosines of these angles are used to define the direction of a vector **a** in 3D space. These are the *direction cosines* of **a** :

dir **a** = (α, β, γ)

α = cos *θ* _{x}

*β* = cos *θ* _{y}

*γ* = cos *θ* _{z}

For any vector **a** in Cartesian three-space, the sum of the squares of the direction cosines is always equal to 1. That is

α ^{2} + *β* ^{2} + *γ* ^{2} = 1

Another way of expressing this is:

cos ^{2} *θ* _{x} + cos ^{2} *θ _{y}* + cos

^{2}

*θ*= 1

_{z}where the expression cos ^{2} *θ* means (cos *θ* ) ^{2}.

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