288 CHAPTER 14. VECTOR PRODUCTS

Example 14.2.1 Find the angle between the vectors 2i+j−k and 3i+4j+k.

The dot product of these two vectors equals 6+4−1 = 9 and the norms are√

4+1+1 =√

6

and√

9+16+1 =√

26. Therefore, from 14.12 the cosine of the included angle equals

cosθ =9√

26√

6= .72058

Now the cosine is known, the angle can be determines by solving the equation cosθ = .72058. This will involve using a calculator or a table of trigonometric functions. The an-swer is θ = .76616 radians or in terms of degrees, θ = .76616× 360

2π= 43.898◦. Recall

how this last computation is done. Set up a proportion x.76616 = 360

2πbecause 360◦ corre-

sponds to 2π radians. However, in calculus, you should get used to thinking in terms ofradians and not degrees. This is because all the important calculus formulas are defined interms of radians.

Example 14.2.2 Let u,v be two vectors whose magnitudes are equal to 3 and 4 respec-tively and such that if they are placed in standard position with their tails at the origin,the angle between u and the positive x axis equals 30◦ and the angle between v and thepositive x axis is -30◦. Find u ·v.

From the geometric description of the dot product in 14.12

u ·v = 3×4× cos(60◦) = 3×4×1/2 = 6.

Observation 14.2.3 Two vectors are said to be perpendicular if the included angle isπ/2 radians (90◦). You can tell if two nonzero vectors are perpendicular by simply takingtheir dot product. If the answer is zero, this means they are perpendicular because cosθ =0.

Example 14.2.4 Determine whether the two vectors 2i+j−k and 1i+3j+5k are per-pendicular.

When you take this dot product you get 2+ 3− 5 = 0 and so these two are indeedperpendicular.

Definition 14.2.5 When two lines intersect, the angle between the two lines is thesmaller of the two angles determined.

Example 14.2.6 Find the angle between the two lines, (1,2,0)+ t (1,2,3) and (0,4,−3)+t (−1,2,−3) .

These two lines intersect, when t = 0 in the first and t = −1 in the second. It is onlya matter of finding the angle between the direction vectors. One angle determined is givenby

cosθ =−614

=−37

. (14.13)

We don’t want this angle because it is obtuse. The angle desired is the acute angle given by

cosθ =37.

It is obtained by using replacing one of the direction vectors with −1 times it.