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Linear Algebra · Vectors & Matrices

Vector Operations — Practice Problems and Worked Solutions

Combine vectors and read coordinates with confidence.

The formulas you need here

Vector length

∣ v ∣ = v \cdot v = v_{1}^{2} + \dots + v_{n}^{2}

What it says: Pythagoras, in as many dimensions as you like.
When to reach for it: Normalising a vector, or any distance in space.
Watch out: A unit vector is v divided by its length — dividing by the length squared is the standard slip.

∣ (3, 4) ∣ = 5

So the unit vector in that direction is (3/5, 4/5).

17 worked problems

Every step is generated and checked by a computer algebra system.

1.
Compute the linear combination.
$1 u + (1) v, u = [12], v = [3 - 1]$
Answer: $[41]$
Step-by-step solution
1. $1 [12] + (1) [3 - 1]$
2. $= [41]$
2.
Compute the linear combination.
$1 u + (1) v, u = [- 2 4], v = [52]$
Answer: $[36]$
Step-by-step solution
1. $1 [- 2 4] + (1) [52]$
2. $= [36]$
3.
Compute the linear combination.
$2 u + (- 1) v, u = [30], v = [- 1 6]$
Answer: $[7 - 6]$
Step-by-step solution
1. $2 [30] + (- 1) [- 1 6]$
2. $= [7 - 6]$
4.
Compute the linear combination.
$1 u + (2) v, u = 2 - 3 1, v = 14 - 2$
Answer: $45 - 3$
Step-by-step solution
1. $1 2 - 3 1 + (2) 14 - 2$
2. $= 45 - 3$
5.
Compute the linear combination.
$3 u + (- 1) v, u = - 1 25, v = 30 - 1$
Answer: $- 6 616$
Step-by-step solution
1. $3 - 1 25 + (- 1) 30 - 1$
2. $= - 6 616$
6.
Compute the linear combination.
$- 1 u + (2) v, u = 4 - 2 1, v = - 2 35$
Answer: $- 8 89$
Step-by-step solution
1. $- 1 4 - 2 1 + (2) - 2 35$
2. $= - 8 89$
7.
Compute the linear combination.
$2 u + (1) v, u = [25], v = [1 - 3]$
Answer: $[57]$
Step-by-step solution
1. $2 [25] + (1) [1 - 3]$
2. $= [57]$
8.
Compute the linear combination.
$3 u + (- 2) v, u = [- 1 3], v = [42]$
Answer: $[- 11 5]$
Step-by-step solution
1. $3 [- 1 3] + (- 2) [42]$
2. $= [- 11 5]$
9.
Compute the linear combination.
$1 u + (3) v, u = [51], v = [- 2 4]$
Answer: $[- 1 13]$
Step-by-step solution
1. $1 [51] + (3) [- 2 4]$
2. $= [- 1 13]$
10.
Compute the linear combination.
$2 u + (3) v, u = 1 - 2 3, v = 21 - 1$
Answer: $8 - 1 3$
Step-by-step solution
1. $2 1 - 2 3 + (3) 21 - 1$
2. $= 8 - 1 3$
11.
Compute the linear combination.
$- 2 u + (1) v, u = 31 - 2, v = - 1 42$
Answer: $- 7 26$
Step-by-step solution
1. $- 2 31 - 2 + (1) - 1 42$
2. $= - 7 26$
12.
Compute the linear combination.
$4 u + (- 1) v, u = 204, v = 13 - 1$
Answer: $7 - 3 17$
Step-by-step solution
1. $4204 + (- 1) 13 - 1$
2. $= 7 - 3 17$
13.
Compute the linear combination.
$- 1 u + (2) v, u = [6 - 1], v = [23]$
Answer: $[- 2 7]$
Step-by-step solution
1. $- 1 [6 - 1] + (2) [23]$
2. $= [- 2 7]$
14.
Compute the linear combination.
$3 u + (2) v, u = 111, v = 2 - 1 3$
Answer: $719$
Step-by-step solution
1. $3111 + (2) 2 - 1 3$
2. $= 719$
15.
Compute the linear combination.
$2 u + (4) v, u = [- 3 2], v = [15]$
Answer: $[- 2 24]$
Step-by-step solution
1. $2 [- 3 2] + (4) [15]$
2. $= [- 2 24]$
16.
Compute the linear combination.
$1 u + (- 3) v, u = 43 - 2, v = 1 - 1 2$
Answer: $16 - 8$
Step-by-step solution
1. $1 43 - 2 + (- 3) 1 - 1 2$
2. $= 16 - 8$
17.
Compute the linear combination.
$3 u + (1) v, u = [27], v = [- 3 1]$
Answer: $[322]$
Step-by-step solution
1. $3 [27] + (1) [- 3 1]$
2. $= [322]$

The formulas you need here

Vector length

17 worked problems

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