Exercise 5.4

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5 . 4 T H E C R O S S P R O D U C T O F T W O V E C T O R S 185

These properties can be checked by using the definition of the cross product.

Notice that the first property means that the cross product is not commutative.

For example,i៬j៬k៮៬,but j៬i៬ k៮៬.

Since the result of a cross product is a vector, you may form the dot product or the cross product of this vector with a third vector. The quantity (u៮៬v៮៬) • w៮៬is known as the triple scalar product of three vectors, because it is a scalar quantity.

The brackets are not really needed to specify the order of operations, because u៮៬(v៮៬• w៮៬) is meaningless. (Why?) The quantity (u៮៬v៮៬)w៮៬is a vector and is called the triple vector product. Brackets are required in this expression to specify the order of operations. Both of these quantities arise in the application of vectors to physical and geometrical problems. Some of their properties are investigated in the exercises.

Part A

1. If w៮៬u៮៬v៮៬, explain why w៮៬• u៮៬,w៮៬• v៮៬, and w៮៬• (au៮៬ bv៮៬) are all zero.

2. Find u៮៬v៮៬for each of the following pairs of vectors. State whether u៮៬v៮៬

is directed into or out of the page.

a. b. c.

3. State whether the following expressions are vectors, scalars, or meaningless.

a. a៮៬• (b៮៬c៮៬) b. (a៮៬• b៮៬)(b៮៬•c៮៬) c. (a៮៬b៮៬) •c៮៬

d. a៮៬(b៮៬•c៮៬) e. (a៮៬b៮៬) • (b៮៬c៮៬) f. (a៮៬b៮៬)c៮៬

g. a៮៬• (b៮៬•c៮៬) h. (a៮៬b៮៬)(b៮៬c៮៬) i. (a៮៬b៮៬)c៮៬

j. a៮៬(b៮៬c៮៬) k. (a៮៬• b៮៬)(b៮៬•c៮៬) l. (a៮៬• b៮៬)c៮៬

4. Use the cross product to find a vector perpendicular to each of the following pairs of vectors. Check your answer using the dot product.

a. (4, 0, 0) and (0, 0, 4) b. (1, 2, 1) and (6, 0, 6) c. (2,1, 3) and (1, 4,2) d. (0, 2,5) and (4, 9, 0) Part B

5. Find a unit vector perpendicular to a៮៬(4,3, 1) and b៮៬(2, 3,1).

Knowledge/

Understanding Communication

u 12°

= 5 v = 3 u

120°

= 18

v = 25 u

68°

= 12 v = 5

Knowledge/

Understanding Communication

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C H A P T E R 5 186

6. Find two vectors perpendicular to both (3,6, 3) and (2, 4, 2).

7. Express the unit vectors iˆ,jˆ, andkˆ as ordered triples and show that

a. iˆjˆkˆ. b. kˆjˆ iˆ.

8. Using components, show that

a. u៮៬v៮៬ v៮៬ u៮៬for any vectors u៮៬andv៮៬. b. u៮៬v៮៬0៮៬, if u៮៬andv៮៬are collinear.

9. Prove that a៮៬b៮៬= 兹(a苶៮៬• a៮៬)(b苶៮៬• b៮៬)苶(a៮៬•苶b៮៬)².

10. Given a៮៬(2, 1, 0),b៮៬(1, 0, 3), and c៮៬(4,1, 1), calculate the following triple scalar and triple vector products.

a. a៮៬b៮៬• c៮៬ b. b៮៬c៮៬• a៮៬ c. c៮៬a៮៬• b៮៬

d. (a៮៬b៮៬)c៮៬ e. (b៮៬c៮៬)a៮៬ f. (c៮៬a៮៬)b៮៬

11. By choosing u៮៬v៮៬, show that u៮៬(v៮៬w៮៬) (u៮៬v៮៬)w៮៬. This means that, in general, the cross product is not associative.

12. Given two non-collinear vectors a៮៬and b៮៬, show that a៮៬,a៮៬b៮៬, and (a៮៬b៮៬)a៮៬

are mutually perpendicular.

13. Prove that the triple scalar product of the vectors u៮៬,v៮៬, and w៮៬has the property that u៮៬• (v៮៬w៮៬)(u៮៬v៮៬) • w៮៬. Carry out the proof by expressing both sides of the equation in terms of components of the vectors.

Part C

14. If the cross product of a៮៬and b៮៬is equal to the cross product of a៮៬and c៮៬, this does not necessarily mean that b៮៬equals c៮៬. Show why this is so

a. by making an algebraic argument.

b. by drawing a geometrical diagram.

15. a.If a៮៬(1, 3,1),b៮៬(2, 1, 5),v៮៬(3,y,z), and a៮៬v៮៬b៮៬, find yand z.

b. Find anothervector v៮៬for which a៮៬v៮៬b៮៬.

c. Explain why there are infinitely many vectors v៮៬for which a៮៬v៮៬b៮៬.

Thinking/Inquiry Problem Solving Communication Thinking/Inquiry Problem Solving Application

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