Page 167 - Advanced engineering mathematics

VECTORS IN THE PLANE AND
3-SPACE THE DOT PRODUCT THE CROSS
CHAPTER 6 PRODUCT THE VECTOR SPACE R N
ORTHOGONALIZATION
Vectors and

Vector Spaces

6.1 Vectors in the Plane and 3-Space

Some quantities, such as temperature and mass, are completely specified by a number. Such
quantities are called scalars. By contrast, a vector has both a magnitude and a sense of direction.
If we push against an object, the effect is determined not only by the strength of the push, but its
direction. Velocity and acceleration are also vectors.
We can include both both magnitude and direction in one package by representing a vector
as an arrow from the origin to a point (x, y, z) in 3-space, as in Figure 6.1. The choice of the point
gives the direction of the vector (when viewed from the origin), and the length is its magnitude.
The greater the force, the longer the arrow representation.

To distinguish when we are thinking of a point as a vector (arrow from the origin to the
point), we will denote this vector < x, y, z >. We call x the first component of < x, y, z >,
y the second component and z the third component. These components are scalars.

Two vectors are equal exactly when their respective components are equal. That is,
< x 1 , y 1 , z 1 >=< x 2 , y 2 , z 2 >
exactly when x 1 = x 2 , y 1 = y 2 , and z 1 = z 2 .
Since only direction and magnitude are important in specifying a vector, any arrow of the
same length and orientation denotes the same vector. The arrows in Figure 6.2 represent the same
vector.
The vector <−x,−y,−z > is opposite in direction to <x, y, z >, as suggested in Figure 6.3.
It is convenient to denote vectors by bold-face letters (such as F,G, and H) and scalars (real
numbers) in ordinary type.

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October 14, 2010 14:21 THM/NEIL Page-147 27410_06_ch06_p145-186