Vectors in Cartesian Three-Space Help

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.)

Fig. 9-7 . Vectors in xyz -space are added using the “parallelogram method.” This is a perspective drawing, so the parallelogram appears distorted.

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 _za ). Then the direction angles of k a are the same; they are also ( θ _xa , θ _ya , θ _za ). The direction angles of − 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). 

Fig. 9-8 . The vector b × a has the same magnitude as vector a × b , but points in the opposite direction. Both cross products are perpendicular to the plane containing the two original vectors.

Unit Vectors

Any vector a, reduced to standard form so its starting point is at the origin, ends up at some point ( x _a , y _a , z _a ). This vector can be broken down into the sum of three mutually perpendicular vectors, each of which lies along one of the coordinate axes as shown in Fig. 9-9:

a = ( x _a , y _a , z _a )

= ( x _a ,0,0) + (0, y _a ,0) + (0,0, z _a )

= x _a (1,0,0) + y _a (0,1,0) + z _a (0,0,1)

The vectors (1,0,0), (0,1,0), and (0,0,1) are called unit vectors because their length is 1. It is customary to name these vectors i , j , and k , because they come in handy:

(1,0,0) = i

(0,1,0) = j

(0,0,1) = k

Therefore, the vector a shown in Fig. 9-9 breaks down this way:

a = ( x _a , y _a , z _a ) = x _a i + y _a j + z _a k

Fig. 9-9 . Any vector in Cartesian three-space can be broken up into a sum of three component vectors, each of which lies on one of the coordinate axes.

PROBLEM 9-6

Break the vector b = (−2, 3, −7) down into a sum of multiples of the unit vectors i , j , and k .

SOLUTION 9-6

This is a simple process—almost trivial—but envisioning it requires a keen “mind’s eye.” If you have any trouble seeing this in your imagination, think of i as “one unit of width going to the right,” j as “one unit of height going up,” and k as “one unit of depth coming towards you.” Here we go:

b = (−2,3, − 7)

= −2 × (1,0,0) + 3 × (0,1,0) + [−7 × (0,0,1)]

= −2 i + 3 j + (−7) k

= −2 i + 3 j − 7 k

Practice problems for these concepts can be found at:  Vectors And Cartesian Three-Space Practice Test.