MHR • Calculus and Vectors 12 Solutions 574 Chapter 6 Section 2 ...

Chapter 6 Section 2:
Addition and Subtraction of Vectors
Chapter 6 Section 2 Question 1 Page 325
a)
b)
c)
d)
Chapter 6 Section 2 Question 2 Page 325
a) First measure the length of the path.
d = 1+1+1.5+ 2 +1.5+1+1+ 2.5+1.5+1.5+1+ 2.5+1.5+1+1+1.5
= 23
Since the scale is 1 cm : 3 m, the actual distance is 23 × 3 or 69 m.
The displacement is represented by a vector from the start (tail) of the first vector to the end (head) of
the last vector. This measures 5 cm in length. Therefore, the actual displacement is 15 m to the right.
b) The distance and the displacement are different. The distance represents the total distance travelled, 69
m. The displacement represents the straight line distance between the initial and final positions, 15 m.
Also, the displacement, a vector quantity, has a direction implied while the distance has no direction
since it is a scalar quantity.
MHR • Calculus and Vectors 12 Solutions 574

Chapter 6 Section 2 Question 3 Page 325
Answers may vary. For example:
a) The shortest vector is w !" !" " "
; w = u ! v
!!!" !!!" !!!" !!!"
b) The shortest vector is AB ; AB=AC ! BC
c) The shortest vector is RQ
!!!" !!!" !!!" !!!"
; RQ = PQ ! PR
d) The shortest vector is e ! ! ! "!
; e = v ! f
Chapter 6 Section 2 Question 4 Page 325
!!!"
a) AB
!!!"
b) BD
!!!"
c) AC
!!!"
d) AC
!!!"
e) BC
!!!"
!!!"
f) AB ! DB
(parallelogram method)
(parallelogram method)
!!!"
= AB
!!!"
+ BD
!!!"
= AD
!!!"
!!!"
!!!"
!!!"
!!!"
!!!"
g) AB ! CB ! DC = AB + BC + CD
!!!"
= AD
h) The vectors form a triangle.
!!!"
!!!
" !!!"
!!!"
!!!"
!!!"
AE ! EB ! BC = AE + BE + CB
!!!"
!!!"
!!!"
= AE + ED + DA
"
= 0
Chapter 6 Section 2 Question 5 Page 326
Since the hexagon is regular, all sides are equal and all interior angles are equal. Consequently opposite
sides are equal and parallel. Also, the interior triangles formed with the centre are equilateral triangles.
!!!"
!!!"
!!!"
AB = AO + OB
!!!"
!!!"
= !OA + OB
!!!"
!!!"
= OB ! OA
" "
= b ! a
MHR • Calculus and Vectors 12 Solutions 575

!!!"
OC
!!!"
CO
!!!"
AE
!!!"
= AB
"
"
= b ! a
!!!"
= !OC
"
"
= !(b ! a)
" "
= !b + a
" "
= a ! b
!!!"
!!!"
= AO + OE
!!!"
!!!"
= !OA ! OB
" "
= !a ! b
Chapter 6 Section 2 Question 6 Page 326
a)
b)
c)
MHR • Calculus and Vectors 12 Solutions 576

Chapter 6 Section 2 Question 7 Page 326
Answers may vary. For example:
a)
b)
Use The Geometer’s Sketchpad® to draw a scale diagram. Use a scale of 1 cm : 33.3 N. Let O
!!"
represent the knot. The resultant vector is OJ . The magnitude of the resultant force on the knot is 17.3
N. The direction of the resultant force on the knot is upward if Allen’s force is directly to the left
(000°).
It is also possible to solve this problem using trigonometry involving the sine law, solving triangle
OAN first and then triangle NOJ.
MHR • Calculus and Vectors 12 Solutions 577