Exam (elaborations)
MAT 1503_assignment_6_2022
- Course
- MAT1503
- Institution
- University Of Johannesburg (UJ)
MAT 1503_assignment_6_2022
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MAT 1503 Assignment 6 2022MAT1503
ASSIGNMENT 6
2023
Linear Algebra
, MAT1503
ASSIGNMENT 6
QUESTION 1
QUESTION 1.1
𝑢⃗ =〈1,3, −2〉 𝑎𝑛𝑑 𝑣 = 〈−5,3,2〉
𝐿𝑒𝑡 𝑡ℎ𝑒 𝑎𝑛𝑙𝑔𝑙𝑒 = 𝜃
𝑢⃗ ∙ 𝑣
cos(𝜃) =
‖𝑢⃗ ‖‖𝑣‖
𝑢⃗ ∙ 𝑣 =〈1,3, −2〉 ∙ 〈−5,3,2〉
𝑢⃗ ∙ 𝑣 = −5 + 9 − 4 = 0
𝑆𝑖𝑛𝑐𝑒 𝑢⃗ ∙ 𝑣 = 0 𝑡ℎ𝑒𝑛 𝑢⃗ 𝑎𝑛𝑑 𝑣 𝑎𝑟𝑒 𝑜𝑟𝑡ℎ𝑜𝑔𝑜𝑛𝑎𝑙 𝑣𝑒𝑐𝑡𝑜𝑟𝑠
QUESTION 1.2
𝑢⃗ =〈1, −2,4〉 𝑎𝑛𝑑 𝑣 = 〈5,3,7〉
𝐿𝑒𝑡 𝑡ℎ𝑒 𝑎𝑛𝑙𝑔𝑙𝑒 = 𝜃
𝑢⃗ ∙ 𝑣
cos(𝜃) =
‖𝑢⃗ ‖‖𝑣‖
𝑢⃗ ∙ 𝑣 =〈1, −2,4〉 ∙ 〈5,3,7〉
= 5 − 6 + 28
= 27
‖𝑢⃗ ‖ = √(1)2 + (−2)2 + (4)2 = √21
‖𝑣‖ = √(5)2 + (3)2 + (2)2 = √38
, 27
cos(𝜃) =
√21 ∙ √38
, 27
𝜃 = cos −1 ( )
√21 ∙ √38
𝜃 = 17.1° 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑖𝑠 𝑎𝑛 𝑎𝑐𝑢⃗𝑡𝑒
QUESTION 2
QUESTION 2.1
𝑎 ∙ 𝑢⃗
𝑝𝑟𝑜𝑎 𝑢⃗ = 2 𝑎
|𝑎|
𝑎 = 〈−1, −3〉
|𝑎| = √(−1)2 + (−3)2 = √10
𝑢⃗ =〈−1,3〉
〈−1, −3〉 ∙ 〈−1,3〉
𝑝𝑟𝑜𝑎 𝑢⃗ = 2 〈−1, −3〉
(√10)
1−9
= 〈−1, −3〉
10
8
=− 〈−1, −3〉
10
4
= − 〈−1, −3〉
5
4 12
=〈 , 〉
5 5
4 12
𝑝𝑟𝑜𝑎 𝑢⃗ =〈 , 〉
5 5
QUESTION 2.2
𝑎 ∙ 𝑢⃗
𝑝𝑟𝑜𝑎 𝑢⃗ = 2 𝑎
|𝑎|
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