Vectors in 3D Help (page 2)
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
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:
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).
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 :
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.
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.
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
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°.
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 )
A Point Of Confusion
Are you confused here about the concept of vector magnitude, and the fact that absolute-value symbols (the two vertical lines) are used to denote magnitude? The absolute value of a number is always positive, but with vectors, negative magnitudes sometimes appear in the equations.
Whenever we see a vector whose magnitude is negative, it is the equivalent of a positive vector pointing in the opposite direction. For example, if a force of –20 newtons is exerted upward, it is the equivalent of a force of 20 newtons exerted downward. When a vector with negative magnitude occurs in the final answer to a problem, you can reverse the direction of the vector, and assign to it a positive magnitude that is equal to the absolute value of the negative magnitude.
Vectors in 3D Practice Problems
What is the magnitude of the vector denoted by a = ( x a , y a , z a ) = (1,2,3)? Consider the values 1, 2, and 3 to be exact; express the answer to four decimal places.
Use the distance formula for a vector in Cartesian xyz -space:
Consider two vectors a and b in xyz -space, both of which lie in the xy -plane. The vectors are represented by the following ordered triples:
a = (3,4,0)
b = (0, –5,0)
Find the ordered triple that represents the vector a × b .
Let’s draw these two vectors as they appear in the xy -plane. See Fig. 6-12. In this drawing, imagine the positive z axis coming out of the page directly toward you, and the negative z axis pointing straight away from you on the other side of the page.
First, let’s figure out the direction in which a × b points. The direction of the cross product of two vectors is always perpendicular to the plane containing the original vectors. Thus, a × b points along the z axis. The ordered triple must be in the form (0,0, z ), where z is some real number. We don’t yet know what this number is, and we had better not jump to any conclusions. Is it positive? Negative? Zero? We must proceed further to find out.
Next, we calculate the lengths (magnitudes) of the two vectors a and b . To find | a |, we use the formula:
Similarly, for | b |:
Therefore, | a | | b | = 5 × 5 = 25. In order to determine the magnitude of a × b , we must multiply this by the sine of the angle θ between the two vectors, as expressed counterclockwise from a to b .
To find the measure of θ , note that it is equal to 270° (three-quarters of a circle) minus the angle between the x axis and the vector a . The angle between the x axis and vector a is the arctangent of 4/3, or approximately 53° as determined using a calculator. Therefore:
θ = 270° – 53° = 217°
sin θ = sin 217° = –0.60 (approx.)
This means that the magnitude of a × b is equal to approximately 25 × (–0.60), or –15. The minus sign is significant. It means that the cross product vector points negatively along the z axis. Therefore, the z coordinate of a × b is equal to –15. We know that the x and y coordinates of a × b are both equal to 0 because a × b lies along the z axis. It follows that a × b = (0,0,–15).
Practice problems for these concepts can be found at: Three-Space and Vectors Practice Test
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