Question

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(b) it is given that p(4; 8) and r(-4; -2) are points on the cartesian plane. find (i) $overrightarrow{pr}$ as a column vector, (ii) $|overrightarrow{pr}|$.

Explanation:

Step1: Find the components of the column - vector

To find the column - vector $\overrightarrow{PR}$, subtract the coordinates of point $P$ from the coordinates of point $R$. If $P(x_1,y_1)=(4,8)$ and $R(x_2,y_2)=(-4,-2)$, then $\overrightarrow{PR}=

$$\begin{pmatrix}x_2 - x_1\\y_2 - y_1\end{pmatrix}$$

=

$$\begin{pmatrix}-4 - 4\\-2 - 8\end{pmatrix}$$

=

$$\begin{pmatrix}-8\\-10\end{pmatrix}$$

$.

Step2: Calculate the magnitude of the vector

The magnitude of a two - dimensional vector $\vec{v}=

$$\begin{pmatrix}a\\b\end{pmatrix}$$

$ is given by $|\vec{v}|=\sqrt{a^{2}+b^{2}}$. For $\overrightarrow{PR}=

$$\begin{pmatrix}-8\\-10\end{pmatrix}$$

$, $|\overrightarrow{PR}|=\sqrt{(-8)^{2}+(-10)^{2}}=\sqrt{64 + 100}=\sqrt{164}=2\sqrt{41}$.

Answer:

(i)

$$\begin{pmatrix}-8\\-10\end{pmatrix}$$

(ii) $2\sqrt{41}$