MAT2611 Assignment 07 (Accurate Solutions) Due 11 July 2025

MAT2611 Assignment 07 Solutions
Student Name: [Your Name]
Due: Friday, 11 July 2025


Problem 1
Problem Statement.
Suppose that u, v, w are vectors such that:


< u, v >= 2, < v, w >= 2, < u, w >= 4, ∥u∥ = 1, ∥v∥ = 3, ∥w∥ = 5.

Evaluate:

(a) < u + v, v + w >

(b) < 2v − w, 3u + 2w >
(c) ∥2u + v∥
(d) d(2v, w)
(e) ∥u − 2v + w∥

Part (a)
Step 1. Expand using linearity:


< u + v, v + w >=< u, v > + < u, w > + < v, v > + < v, w > .

Step 2. Substitute values:

= 2 + 4 + 9 + 2 = 17.
Final Answer.

< u + v, v + w >= 17.
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, Part (b)
Step 1. Expand:


< 2v − w, 3u + 2w >= 6 < u, v > +4 < v, w > −3 < u, w > −2 < w, w > .

Step 2. Substitute:

= 6(2) + 4(2) − 3(4) − 2(25),


= 12 + 8 − 12 − 50,


= −42.
Final Answer.

< 2v − w, 3u + 2w >= −42.
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Part (c)
Step 1. Compute:

∥2u + v∥2 =< 2u + v, 2u + v >


= 4 < u, u > +4 < u, v > + < v, v > .
Step 2. Substitute:

= 4(1) + 4(2) + 9,


= 4 + 8 + 9 = 21.
Final Answer.
√
∥2u + v∥ = 21.
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