Vectors - Algebraic Operations

Maths: Edexcel A Level Pure Maths, topic 10.1.3

Adding and Subtracting Vectors

Writing vectors in component form makes adding and subtracting vectors very simple.

If we have two vectors $\mathbf{a}$ and $\mathbf{b}$ , we can write their sum as:
$\mathbf{a} + \mathbf{b} = (a_i\mathbf{i} + a_j\mathbf{j}) + (b_i\mathbf{i} + b_j\mathbf{j}) = (a_i + b_i)\mathbf{i} + (a_j + b_j)\mathbf{j}$
Where $a_i,b_i$ are the $\mathbf{i}$ components of $\mathbf{a},\mathbf{b}$ , and likewise for $\mathbf{j}$ .
In column form this is:
$\mathbf{a} + \mathbf{b} = {a_i\choose a_j}+{b_i \choose b_j} = {a_i + \,b_i \choose a_j+\,b_j}$

If we have two vectors $\mathbf{a}$ and $\mathbf{b}$ , we can write their difference as:
$\mathbf{a} - \mathbf{b} = (a_i\mathbf{i} + a_j\mathbf{j}) - (b_i\mathbf{i} + b_j\mathbf{j}) = (a_i - b_i)\mathbf{i} + (a_j - b_j)\mathbf{j}$
In column form this is:
$\mathbf{a} - \mathbf{b} = {a_i\choose a_j}-{b_i \choose b_j} = {a_i -\, b_i \choose a_j-\,b_j}$

What is the $\mathbf{i}$ -component of $\mathbf{t} + \mathbf{p}$ ?
$\mathbf{t} = 4\mathbf{i} + 5\mathbf{j}$ and $\mathbf{p} = 5\mathbf{i} - 3\mathbf{j}$
What is the $\mathbf{i}$ -component of $\mathbf{t} + \mathbf{p}$ ?
$\mathbf{t} = {6\choose1}$ and $\mathbf{p} = 5\mathbf{i} - 3\mathbf{j}$
What is the $\mathbf{i}$ -component of $\mathbf{t} - \mathbf{p}$ ?
$\mathbf{t} = 5\mathbf{i} - 3\mathbf{j}$ and $\mathbf{p} = {4\choose12}$
What is the $\mathbf{j}$ -component of $\mathbf{t} - \mathbf{p}$ ?
$\mathbf{t} = -\mathbf{i} + 9\mathbf{j}$ and $\mathbf{p} = 7\mathbf{i} - 2\mathbf{j}$