**Introduction**

A vector has two independently variable properties: magnitude and direction. Vectors are used commonly in physics to represent phenomena such as force, velocity, and acceleration. In contrast, real numbers, also called *scalars* , are one-dimensional (they can be depicted on a line); they have only magnitude. Scalars are satisfactory for representing phenomena or quantities such as temperature, time, and mass.

**Vectors In Two Dimensions**

Do you remember *rectangular coordinates* , the familiar *xy* plane from your high-school algebra courses? Sometimes this is called the *cartesian plane* (named after the mathematician Rene Descartes.) Imagine two vectors in that plane. Call them **a** and **b** . (Vectors are customarily written in boldface, as opposed to variables, constants, and coefficients, which are usually written in italics). These two vectors can be denoted as rays from the origin (0, 0) to points in the plane. A simplified rendition of this is shown in Fig. 1-6.

**Fig. 1-6** . Vectors in the rectangular *xy* plane.

Suppose that the end point of **a** has values ( *x _{a}* ,

*y*) and the end point of

_{a}**b**has values (

*x*,

_{b}*y*). The magnitude of

_{b}**a**, written |

**a**|, is given by

The sum of vectors **a** and **b** is

**a** + **b** = [( *x _{a}* +

*x*), (

_{b}*y*+

_{a}*y*)]

_{b}This sum can be found geometrically by constructing a parallelogram with **a** and **b** as adjacent sides; then **a** + **b** is the diagonal of this parallelogram.

The *dot product* , also known as the *scalar product* and written **a** · **b** , of vectors **a** and **b** is a real number given by the formula

**a · b** = *x _{a} x _{b}* +

*y*

_{a}y_{b}The *cross product* , also known as the *vector product* and written **a** × **b** , of vectors **a** and **b** is a vector perpendicular to the plane containing **a** and **b** . Suppose that the angle between vectors **a** and **b** , as measured counterclockwise (from your point of view) in the plane containing them both, is called *q* . Then **a** × **b** points toward you, and its magnitude is given by the formula

| **a** × **b** | = | **a** | | **b** | sin *q*

**Vectors In Three Dimensions**

Now expand your mind into three dimensions. In rectangular *xyz* space, also called *cartesian three-space* , two vectors **a** and **b** can be denoted as rays from the origin (0, 0, 0) to points in space. A simplified illustration of this is shown in Fig. 1-7.

**Fig. 1-7** . Vectors in three-dimensional *xyz* space.

Suppose that the end point of a has values ( *x _{a}* ,

*y*,

_{a}*z*) and the end point of b has values (

_{a}*x*,

_{b}*y*,

_{b}*z*). The magnitude of

_{b}**a**, written |

**a**|, is

The sum of vectors **a** and **b** is

**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.

The dot product **a** · **b** of two vectors **a** and **b** in *xyz* space is a real number given by the formula

**a** · **b** = *x _{a} x _{b}* +

*y*+

_{a}y_{b}*z*

_{a}z_{b}

The cross product **a** × **b** of vectors **a** and **b** in *xyz* space is a little more complicated to envision. It is a vector perpendicular to the plane *P* containing both **a** and **b** and whose magnitude is given by the formula

| **a** × **b** | = | **a** | | **b** | sin *q*

where sin *q* is the sine of the angle *q* between **a** and **b** as measured in *P* . The direction of the vector **a** × **b** is perpendicular to plane *P* . If you look at a and b from some point on a line perpendicular to *P* and intersecting *P* at the origin, and *q* is measured counterclockwise from **a** to **b** , then the vector **a** × **b** points toward you.

Practice problems for these concepts can be found at: Equations, Formulas, And Vectors for Practice Test

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