Work through the following questions to ensure you know how to add and subtract vectors in 2D.
\nFor the whole of this question:
\n$\\bf{a} = \\var{a}$ and $\\bf{b}=\\var{b}$.
", "advice": "The vectors in this question have two dimensions but the idea of addition and subtraction of vectors works in any number of dimensions (as long as all the vectors being added or subtracted have the same dimensions as each other).
\nTo add two vectors you simply add their corresponding elements. In general:
\n$$
\\left(\\begin{array}{c}
a \\\\
b \\\\
\\end{array}\\right) +
\\left(\\begin{array}{c}
c \\\\
d \\\\
\\end{array}\\right) =
\\left(\\begin{array}{c}
a+c \\\\
b+d \\\\
\\end{array}\\right).
$$
Subtraction works in the same way so we have:
\n1)
\n$$
\\var{a} + \\var{b} = \\var{a+b}.
$$
2)
\n$$
\\var{a} - \\var{b} = \\var{a-b}.
$$
In order to undertstand the third part of the question you need to know what a \"position vector\" and \"direction vector\" are.
\nA position vector is defined as a vector that symbolises the location of any given point with respect to the origin. It can be thought of as a coordinate point, but written as a column vector - top entry is the \"x-coordinate\" and the bottome entry is the \"y-coordinate\".
\nA direction vector is defined as a vector that symbolises a direction and a distance in that direction but with no specified \"starting point\". In 2D it can be summarized as an instruction to go the top element number of units left or right based on the sign of the element and the bottom element number of units up or down based on the sign of the element.
\nSo the direction vector from $A$ to $B$ can be worked out by looking at a route from $A$ to $B$ that travels along the position vectors given. Starting at $A$ we have to go backwards down $\\bf{a}$ to the origin and then forwards along $\\bf{b}$. This corresponds to doing \"minus\" $\\bf{a}$ and \"positive\" $\\bf{b}$:
\n3)
\n$$
\\vec{AB} = (-)\\bf{a} + \\bf{b} = \\bf{b}-\\bf{a} = \\var{b}-\\var{a} = \\var{b-a}.
$$
Use this link to find some resources which will help you revise this topic.
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