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# The point (-3,-4) divides the line joining point A(-6,-7) and point B in the ratio 1:3. Find the coordinates of B.

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The point (-3,-4) divides the line joining point A(-6,-7) and point B in the ratio 1:3. Find the coordinates of B.

Jun 29, 2017

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The point (-3,-4) divides the line joining point A(-6,-7) and point B in the ratio 1:3.

Find the coordinates of B.

$$\text{Set } \vec{P} = \binom{-3}{-4} \\ \text{Set } \vec{A} = \binom{-6}{-7} \\ \text{Set } \vec{B} =\ ?$$

Formula:

$$\begin{array}{|rcll|} \hline \vec{P} = (1-\lambda)\vec{A}+\lambda\vec{B} \\ \hline \end{array}$$

Ratio 1:3

$$\begin{array}{|rcll|} \hline \lambda &=& \frac{1}{1+3} \\ &=& \frac14 \\ \hline \end{array}$$

Solution for $$\vec{B}$$:

$$\begin{array}{|rcll|} \hline \vec{P} &=& (1-\lambda)\vec{A}+\lambda\vec{B} \\ \lambda\vec{B} &=& \vec{P} - (1-\lambda)\vec{A} \\ \vec{B} &=& \frac{1}{\lambda} \cdot \Big( \vec{P} - (1-\lambda)\vec{A} \Big) \quad & | \quad \lambda &=& \frac14 \\ \vec{B} &=& \frac{1}{ \frac14 } \cdot \Big( \vec{P} - (1-\frac14)\vec{A} \Big) \\ \vec{B} &=& 4 \cdot ( \vec{P} - \frac34 \vec{A} ) \\ \vec{B} &=& 4 \vec{P} - 3\vec{A} \\ \vec{B} &=& 4 \binom{-3}{-4} - 3\binom{-6}{-7} \\ \vec{B} &=& 4 \binom{-3}{-4} + 3\binom{6}{7} \\ \vec{B} &=& \binom{-12}{-16} + \binom{18}{21} \\ \vec{B} &=& \binom{-12+18}{-16+21} \\ \mathbf{ \vec{B} } & \mathbf{=} & \mathbf{\dbinom{6}{5}} \\ \hline \end{array}$$

Point B(6,5)

Jun 30, 2017