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# Question 1. (a) Two particles bounce off each other and continue moving in different directions. The first particle’s direction is defined by ~v = 6~i−~j+~k and the second particle’s direction is...

Question 1.
(a) Two particles bounce off each other and continue moving in different
directions. The first particle’s direction is defined by ~v = 6~i−~j+~k and
the second particle’s direction is defined by ~w = 2~i + 2~j − ~k. Use the
vector product to find the angle between the paths of the two particles.
(b) Four points A(2, 0, 1), B(−1, 2, 3), C(3, 2, 2) and D(3,−6,−3) are cho-
sen to be the vertices of a 3-dimensional shape. Show that the con-
struction of such a 3-dimensional shape is impossible in this case. [9 marks]
(c) Determine whether the following two planes x + 4y − z = 7 and 5x−
3y− 7z = 11 are parallel, orthogonal, coincident (that is, the same) o
none of these. [6 marks]
[Total: 24
marks]
Question 2.
Let ~u =~i−~j + 2~k, ~v = 2~i− 3~j + ~k.
(a) Calculate the dot product of ~u and ~v. Show your working. [6 marks]
(b) Calculate the vector product of ~u and ~v. Show your working. [6 marks]
[Total: 12
marks]
Question 3.
Given matrix A =
 1 2 52 −1 3
1 1 −1

(a) Write the matrix 4A; [4 marks]
(b) Use matrix multiplication to find A2 = A ·A. Show your working; [6 marks]
(c) Find detA by expanding any column. Show your working. [6 marks]
[Total: 16
marks]
Question 4.
Suppose matrix product AB is defined.
(a) If A is 3 × 6 and B is a column matrix, give the dimensions of B and
AB. [4 marks]
(b) If A is the identity matrix and B is 6 × 6, what size is A? [4 marks]
(c) If A is 3 × 8 and AB is 3 × 7, what size is B? [4 marks]
[Total: 12
marks]
2
Question 5.
Find the determinant of the matrices below by inspection. Give your reason
in each case.
B =

1 7 −4 −5
1 5 7 −4
0 21 −12 −15
5 25 35 −20
 C =
 −1 2 −30 5 7
0 0 9
 D =
 − XXXXXXXXXX
0 0 1
.
[4 marks fo
each case]
[Total: 12
marks]
Question 6.
Find the inverse of the matrix D =
 XXXXXXXXXX
2 0 0
. Show your working. [24 marks]
Total: 100 marks
3
Answered Same Day Jul 14, 2021

## Solution

Himanshu answered on Jul 17 2021
I
I
l
(h)
~e = v.w
\ v\lw\
= J1.-1-l
J 3 b + I -+ I J.---~-+-Lf_-+_I
6 = ~-l ( 3-)
l ITT
A ( 'l , O 1 1) ) B (-- i , a_ , ?, J > C ( 3 , .1.
1
l_) , j) ( 3 , - b
1
- 3 j
AB = 31 + c-1J) + (-.&')
-) I\ l\ I\
Be = _ tf, -'" o} -+- 1 k
cb -- o1 + si ~ 5k
~ .--1
AS X &(;
:=. o + Lt o - Ya :::. o
~: Pi~~ .x.+41r2 CC- 7
Sx. -:30-72 ~ ll
~e = s-, 1 +1 =
J \ + \ b+ t J 25+ 'l -+- 4 ~
~6= 0
.\ t) = 30
0
•90 p~ ~ ~~n.9-1
D
Gu..u:tien lb-2
ln_) :Dot ~
~ ~ • I ru.. V -:::. u.,v, + Ll1..V1.
. = (f-f-\-1:).lii-~J+k)
~ ( \x.1) + (-\x--3)+ L2..x\)
= 2 +3+2.
'::::- 7
l.b) ~ec.t~ ~
7.'-\ r,.. 'k:
~ -i 1
~ -3 i
-1 l.
-3 1
j_ 1 + ~ i -i
1 i l -3
_ t (t-i}i - [-3]xl)-J (, x' -Q.x2)-+ k (\ x[-~1- 1x[-1])
~ 'f (s) ;- !(3)-~(,)

~willetl 3
\Alr\ . ~· A~ i
.1_
1
lo.) y t\ = 4 i
l
i
Lb) !l. A == A.A
: i Q__
!2. -1
i .i
\0
-=
3
::i
lC)...
SOLUTION.PDF