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Vectors in 3D Help (page 2)

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

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

Practice 1

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.

Solution 1

Use the distance formula for a vector in Cartesian xyz -space:

Three-Space and Vectors Vectors in 3D A Point Of Confusion

Practice 2

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 .

Solution 2

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.

Three-Space and Vectors Vectors in 3D A Point Of Confusion

Fig. 6-12. Illustration for Solution 2.

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:

Three-Space and Vectors Vectors in 3D A Point Of Confusion

Similarly, for | b |:

Three-Space and Vectors Vectors in 3D A Point Of Confusion

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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