Post on 19-Feb-2021
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�
&�