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Vectors in 3D Help

By — McGraw-Hill Professional
Updated on Oct 3, 2011

Vectors in 3D

In rectangular xyz -space, vectors a and b can be denoted as rays from the origin (0,0,0) to points ( x a , y a , z a ) and ( x b , y b , z b ) as shown in Fig. 6-9.

Magnitude, Direction, and Sum

Magnitude

The magnitude of a , written | a |, can be found by a three-dimensional extension of the Pythagorean theorem for right triangles. The formula looks like this:

Three-Space and Vectors Vectors in 3D Magnitude

Direction

The direction of a is denoted by measuring the angles θ x , θ y , and θ z that the vector a subtends relative to the positive x , y, and z axes respectively (Fig. 6-10).

Three-Space and Vectors Vectors in 3D Direction

Fig. 6-10. Direction angles of a vector in xyz -space.

These angles, expressed in radians as an ordered triple ( θ x , θ y , θ z ), are

the direction angles of a . Sometimes the cosines of these angles are specified. These are the direction cosines of a :

Three-Space and Vectors Vectors in 3D Direction

Sum

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 determined by the diagonal of the parallelogram, as shown in Fig. 6-9.

Three-Space and Vectors Vectors in 3D Direction

Fig. 6-9. Two vectors a and b in xyz -space. They are added using the “parallelogram method.” This is a perspective drawing, so the parallelogram appears distorted.

Multiplication By Scalar and Dot Product

Multiplication By Scalar

In three-dimensional Cartesian coordinates, let vector a be defined by the coordinates ( x a , y a , z a ). Suppose a is multiplied by some positive real scalar k. Then the following equation holds:

k a = k ( x , 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 ( θ x , θ y , θ z ). The direction angles of ka are also ( θ x , θ y , θ z ). The direction angles of – ka are all increased by 180° (π rad), so they are represented by [( θ x + π), ( θ y + π), ( θ Z + π)]. The same effect can be accomplished by subtracting 180° (π rad) from each of these direction angles.

Dot Product

The dot product, also known as the scalar product and written a · b , of vectors a and b in Cartesian xyz -space is a real number given by the formula:

a · b = x a x b + y a y b + z a z b

where a = ( x a , y a , z a ) and b = ( x b , y b , z b ).

The dot product a · b 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 and b is a vector perpendicular to the plane containing a and b . Let θ be the angle between vectors a and b expressed counterclockwise (as viewed from above, or the direction of the positive z axis) in the plane containing them both (Fig. 6-11). 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 sense in which 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 < θ < 2π), the cross-product vector reverses direction compared with the situation when 0° < θ < 180° (0 < θ < π). This is demonstrated by the fact that, in the above formula, sin θ is positive when 0° < θ < 180° (0 < θ < π ), but negative when 180° < θ < 360° (π < θ < 2π). When 180° < θ < 360° (π < θ < 2π), the right-hand rule doesn’t work. Instead, you must use your left hand, and curl your fingers into almost a complete circle! An example is the cross product b × a in Fig. 6-11. The angle ø, expressed counterclockwise between these vectors (as viewed from above), is more than 180°.

Three-Space and Vectors Vectors in 3D Cross Product

Fig. 6-11. The vector b × a has the same magnitude as vector a × b , but points in the opposite direction. Both vectors b × a and a × b are perpendicular to the plane defined by a and b .

For any two vectors a and b , the vector b × a is a “mirror image” of a × b , where the “mirror” is the plane containing both vectors. One way to imagine the “mirror image” is to consider that b × a has the same magnitude as a × b , but points in exactly the opposite direction. Putting it another way, the direction of b x a is the same as the direction of a × b , but the magnitudes of the two vectors are additive inverses (negatives of each other). The cross product operation is not commutative, but the following relationship holds:

a × b = –( b × a )

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