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Vectors in Cartesian Three-Space Help (page 2)

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By — McGraw-Hill Professional
Updated on Oct 24, 2011

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.)

Vectors and Cartesian Three-Space Vectors in Cartesian Three-Space Sum

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). 

Vectors and Cartesian Three-Space Vectors in Cartesian Three-Space Cross Product

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.

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