**Sum**

The sum of vectors **a** = ( *x* _{a} , *y* _{a} , *z* _{a} ) and **b** = ( *x* _{b} , *y* _{b} , *z* _{b} ) in standard form is given by the following formula:

**a** + **b** = [( *x* _{a} + *x* _{b} ), ( *y* _{a} + *y* _{b} ), ( *z* _{a} + *z* _{b} )]

This sum can, as in the two-dimensional case, be found geometrically by constructing a parallelogram with **a** and **b** as adjacent sides. The sum **a** + **b** is the diagonal of the parallelogram. This is shown in Fig. 9-7. (The parallelogram appears distorted because of the perspective of the drawing.)

**Multiplication By Scalar and The Dot and Cross Product **

**Multiplication By Scalar**

In three-dimensional Cartesian coordinates, let vector **a** be defined by the coordinates ( *x* _{a} , *y* _{a} , *z* _{a} ) when reduced to standard form. Suppose **a** is multiplied by a positive real scalar *k* . Then the following equation holds:

*k* **a** = *k* ( *x* _{a} , *y* _{a} , *z* _{a} ) = ( *kx* _{a} , *ky* _{a} , *kz* _{a} )

If **a** is multiplied by a negative real scalar − *k* , then:

− *k* **a** = − *k* ( *x* _{a} , *y* _{a} , *z* _{a} ) = (− *kx* _{a} , − *ky* _{a} , − *kz* _{a} )

Suppose the direction angles of **a** are represented by the ordered triple ( *θ _{xa}* ,

*θ*,

_{ya}*θ*). Then the direction angles of

_{za}*k*

**a**are the same; they are also (

*θ*,

_{xa}*θ*,

_{ya}*θ*). The direction angles of −

_{za}*k*

**a**are all changed by 180° (π rad). The direction angles of −

*k*

**a**are obtained by adding or subtracting 180° (π rad) to each of the direction angles for

*k*

**a**, so that the results are all at least 0° (0 rad) but less than 360° (2π rad).

**Dot Product**

The *dot product* , also known as the *scalar product* and written **a • b** , of vectors **a** = ( *x* _{a} , *y* _{a} , *z* _{a} ) and **b** = ( *x* _{b} , *y* _{b} , *z* _{b} ) in standard form is a real number given by the formula:

**a • b** = *x* _{a} *x* _{b} + *y* _{a} *y* _{b} + *z* _{a} *z* _{b}

The dot product can also be found from the magnitudes | **a** | and | **b** |, and the angle *θ* between vectors **a** and **b** as measured counterclockwise in the plane containing them both:

**a • b** = | **a** || **b** | cos *θ*

**Cross Product**

The *cross product* , also known as the *vector product* and written **a** × **b** , of vectors **a** = ( *x* _{a} , *y* _{a} , *z* _{a} ) and **b** = ( *x* _{b} , *y* _{b} , *z* _{b} ) in standard form is a vector perpendicular to the plane containing **a** and **b** . Let *θ* be the angle between vectors **a** and **b** as measured counterclockwise in the plane containing them both, as shown in Fig. 9-8. The magnitude of **a** × **b** is given by the formula:

| **a** × **b** | = | **a** || **b** | sin *θ*

In the example shown, **a** × **b** points upward at a right angle to the plane containing both vectors **a** and **b** . If 0° < *θ* < 180° (0 < *θ* < π), you can use the *right-hand rule* to ascertain the direction of **a** × **b** . Curl your fingers in the direction that *θ* , the angle between **a** and **b** , is defined. Extend your thumb. Then **a** × **b** points in the direction of your thumb.

When 180° < *θ* < 360° (π rad < *θ* < 2π rad), the cross-product vector reverses direction because its magnitude becomes negative. This is demonstrated by the fact that, in the above formula, sin *θ* is positive when 0° < *θ* < 180° (0 rad < *θ* < π rad), but negative when 180° < *θ* < 360° (π rad < *θ* < 2π rad).

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