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NSW Y12 Maths - Extension 1 Vectors Vectors in Two Dimensions

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Vectors in Two Dimensions Theory

Position vectors are vectors with their tail at the origin \\ The magnitude of a position vector\\ \(\overrightarrow{O H}=\left(\begin{array}{l} a \\ b \end{array}\right) \text { is }|\overrightarrow{O A}|=\sqrt{a^2+b^2}\)\\ Equal vectors\\ \(\text { If }\left(\begin{array}{l} a \\ b \end{array}\right)=\left(\begin{array}{c} c \\ d \end{array}\right) \text { then } a=c \text { and } b=d\)\\ \begin{multicols}{2} \textbf{Example 1}\\ Find the column vector \(a\) which has an initial point \((-2,3)\) and finishes at \((4,6)\)\\ \textbf{Example 1 solution}\\ $\begin{aligned} \left(\begin{array}{l} a \\ b \end{array}\right) & =\left(\begin{array}{c} 4 \\ 6 \end{array}\right)-\left(\begin{array}{c} -2 \\ 3 \end{array}\right) \\ & =\left(\begin{array}{l} 6 \\ 3 \end{array}\right) \end{aligned}$\\ \columnbreak \textbf{Example 2}\\ If a vector \(d=\left(\begin{array}{c}-6 \\ 10\end{array}\right)\) has a terminal point \(\left(\begin{array}{c}11 \\ -3\end{array}\right)\) determine the coordinates of the initial point \\ \textbf{Example 2 solution}\\ $\begin{aligned} \left(\begin{array}{c} 11 \\ -3 \end{array}\right)-\left(\begin{array}{l} a \\ b \end{array}\right)&=\left(\begin{array}{c} -6 \\ 10 \end{array}\right) \\ \left(\begin{array}{c} 11-a \\ -3-b \end{array}\right)&=\left(\begin{array}{c} -6 \\ 10 \end{array}\right) \\ 11-a&=-6 \rightarrow a=17 \\ -3-b&=10 \rightarrow b=-13 \\ & \therefore(17,-13) \end{aligned}$\\ \end{multicols}

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