Math 234

Pre- Lab 9: Fun with Vectors (Sec 7.1-7.4)

The magnitude of a vector �� � 〈��, ��, … , �〉 is ‖��‖ � ���� ��� ⋯ ��

1.) Given the vectors �� �� 4,1,2 � and ��� �� 3,1,0 �. a.) Compute 2�� � 3���. (Your answer should be a vector.)

b.) Compute �2�� � 3����. (Your answer should be a scalar.)

c.) Compute 2‖��‖ � 3�����. (Note this is different from part b.)

The dot product is the sum of the products of respective components

nnbabababa +++=⋅ Lvv

2211

The cross product is found by computing a 3x3 determinant.

=×

321

321det

bbb

aaa

kji

ba

vvv

vv

The resulting vector bavv

× is perpendicular to both av and b

v

. 2.) Let 1,1,0=a

v and 2,1,2=b

v

.

a.) Compute bavv⋅ and ba

vv× .

b.) Compute the angle between vectors av and b

v

using the formula ba

bavv

vv

cos

⋅=θ . (You should be

able to evaluate the inverse cosine in this example without a calculator.)

c.) Compute the angle between vectors av and b

v

using the formula ba

bavv

vv

sin

×=θ . (You should get

the same answer as in part b.)

We can form a vector indicating the displacement between two points

by subtracting the point coordinates:

�� � ���� !"�#$ � �%#� !"�#$ The magnitude ‖��‖ equals the distance between the two points.

The area of the parallelogram formed by vectors �� and ��� is given by the magnitude of the cross product:

'&(� �&�))()!*&�+ ���� , ���� To find the area of the triangle formed by vectors �� and ���, we just take half of the formula above:

'&(�-&"��*)( � 12 ��� , ����

3.) Find the area of the triangle whose corners are given by the three points �3,2,1$.�2,0,2$/�1,�1,0$ (Hint: First find any two vectors along the sides of the triangle.)

Application: Work The dot product can be used to compute the work done on

an object. If a force 0� is applied to move an object a distance ��, then the work is

1 � 0� ∙ ��

4.) Teddiursa pushes a sleeping Snorlax from the point (1,2,3) to the point (3,4,-2) by applying a

force 0� � 2" 54 � 35. Compute the work done by Teddiursa on the sleeping Snorlax.

Application: Torque The cross product can be used to calculate an object's

resistance to rotation. If a force 0� is applied to the end of a lever with position vector &�, then the torque vector is

6� � &� , 0� 5.) Teddiursa loosens a hex nut on the wall by applying a force of 30 N directly downwards to

the end of wrench held along the vector 3" � 24 55. Compute the torque vector.

��

0�

0�

&�