http://www.ck12.org Chapter 7. Vectors
Scalar multiplicationmeans to multiply a vector by a number. This changes the magnitude of the vector, but not
its direction. If−→v =< 3 , 4 >, then 2−→v=< 6 , 8 >.
Example A
Two vectors−→a and−→b, have magnitudes of 5 and 9 respectively. The angle between the vectors is 53◦. Find
|−→a+−→b|.
Solution: Adding vectors can be done in either order (just like with regular numbers). Subtracting vectors must
be done in a specific order or else the vector will be negative (just like with regular numbers). In either case,
use geometric reasoning and the law of cosines with the parallelogram that is formed to find the magnitude of the
resultant vector.
In order to fine the magnitude of the resulting vector(x), note the triangle on the bottom that has sides 9 and 5 with
included angle 127◦.
x^2 = 92 + 52 − 2 · 9 · 5 ·cos 127◦
x≈ 12. 66Example B
Using the picture from Example A, what is the angle that the sum−→a+−→b makes with−→a?
Solution: Start by drawing a good picture and labeling what you know.|−→a|= 5 ,|−→b|= 9 ,|−→a+−→b|≈ 12 .66. Since
you know three sides of the triangle and you need to find one angle, this is the SSS application of the Law of Cosines.