MAT1503 ASSIGNMENT 3 2025

MAT1503
ASSIGNMENT 3
2024

, QUESTION 1


1.1).


⃗ = 〈1, −3, −1〉
u and a⃗ = 〈2,1, −1〉


⃗ ∙ a⃗ = 〈1, −3, −1〉 ∙ 〈2,1, −1〉
u
= 2−3+1
=0


Since the dot product is zero (u
⃗ ∙ a⃗ = 0) then the vectors u
⃗ and a⃗ are orthogonal
vectors.


1.2).



⃗ = 〈1,0, −2〉
v and ⃗ = 〈2, −1, −1〉
b



⃗ ∙ ⃗b = 〈1,0, −2〉 ∙ 〈2, −1, −1〉
v
= 2+0+2
=4




⃗ ‖ = √(1)2 + (0)2 + (−2)2
‖v

= √1 + 0 + 4

= √5


⃗ ‖ = √(2)2 + (−1)2 + (−1)2
‖b

= √4 + 1 + 1

= √6

, ⃗ ∙ ⃗b
v
cos(θ) =
‖v
⃗ ‖‖b ⃗‖

4
cos(θ) =
√5√6
4
cos(θ) =
√30
4
θ = cos−1 ( )
√30
θ = 0.752° (acute angle)


The vectors are not orthogonal, and they make an acute angle.




1.3).



⃗ ∙ ⃗b
v
projb⃗ v
⃗ = ⃗
2b
⃗‖
‖b
4
proj⃗b v
⃗ = 2
〈2, −1, −1〉
(√6)
4
⃗ = 〈2, −1, −1〉
proj⃗b v
6
2
⃗ = 〈2, −1, −1〉
proj⃗b v
3
4 2 2
⃗ = 〈 ,− ,− 〉
proj⃗b v
3 3 3

, QUESTION 2


2.1).


⃗ = 〈x, y, z〉
n


Given n
⃗ is perpendicular to all three vectors.


⃗ ∙u
n ⃗ =0
〈x, y, z〉 ∙ 〈1,1, −1〉 = 0

x+y−z= 0 1




⃗ ∙v
n ⃗ =0
〈x, y, z〉 ∙ 〈−1,1,0〉 = 0

−x + y = 0 2




⃗ ∙ ⃗w
n ⃗⃗ = 0
〈x, y, z〉 ∙ 〈−1,0,1〉 = 0

−x + z = 0 3


x+y−z=0
Solve { −x + y = 0
−x + z = 0


1 1 −1 0
Augumented matrix = [−1 1 0 | 0]
−1 0 1 0