Line RS passes through points R (5, 3) and S (-1, 0).
Line PQ is parallel to line RS and passes through points P (3, -1) and Q (-2, y).
What are the coordinates of point Q?
+26367 Line RS passes through points R (5, 3) and S (-1, 0).
Line PQ is parallel to line RS and passes through points P (3, -1) and Q (-2, y).
What are the coordinates of point Q ?
$$\small{\text{Line PQ is parallel to line RS - cross product }}
\boxed{|(\vec{S}-\vec{R}) \times (\vec{Q}-\vec{P})| = 0}\\ \\
\left|
\left[
\binom{-1}{0}-\binom{5}{3}
\right]
\times
\left[
\binom{-2}{y}-\binom{3}{-1}
\right]
\right| = 0\\\\
\left|
\binom{-6}{-3} \times \binom{-5}{y+1}
\right| = 0\\\\
(-6)\cdot(y+1)-(-3)\cdot(-5) = 0\\
(-6)\cdot(y+1)-15 = 0\\
(-6)\cdot(y+1)=15\\\\
y+1=-\frac{15}{6}\\\\
y=-\frac{15}{6}-1\\\\
y=-\frac{21}{6}\\\\
y= -\frac{7}{2}\\\\
y=-3.5\\\\
\boxed{\vec{Q}=\binom{-2}{-3.5}}$$
+980 The lines are parralel. This means they have the same gradient.
First we find the gradient of RS.
The formula for gradient is: m = (y{2}-y{1})/(x{2}-x{1}).
y{2}=0, y{1}=3, x{2}=-1, x{1}=5
$${\mathtt{m}} = {\frac{\left({\mathtt{0}}{\mathtt{\,-\,}}{\mathtt{3}}\right)}{\left({\mathtt{\,-\,}}{\mathtt{1}}{\mathtt{\,-\,}}{\mathtt{5}}\right)}}$$
$${\mathtt{m}} = {\frac{\left(-{\mathtt{3}}\right)}{\left(-{\mathtt{6}}\right)}}$$
$${\mathtt{m}} = {\frac{{\mathtt{1}}}{{\mathtt{2}}}}$$
Now we can sub our gradient into the equation for line PQ's gradient.
m = (y{2}-y{1})/(x{2}-x{1})
$${\frac{{\mathtt{1}}}{{\mathtt{2}}}} = {\frac{\left({\mathtt{y}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{\left({\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,-\,}}{\mathtt{3}}\right)}}$$ (keep in mind the y corresponds to point Q)
$${\frac{{\mathtt{1}}}{{\mathtt{2}}}} = {\frac{\left({\mathtt{y}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{\left(-{\mathtt{5}}\right)}}$$ Now we solve for y.
$$-{\mathtt{5}} = {\mathtt{2}}{\mathtt{\,\times\,}}\left({\mathtt{y}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)$$
$$-{\mathtt{5}} = {\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{y}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}$$
$$-{\mathtt{7}} = {\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{y}}$$
$${\mathtt{y}} = -{\mathtt{3.5}}$$
We can check this by substituting y into the equation. Does (-3.5 --1)/(-5) = 1/2?
This is our y coordinate for point Q.
So point Q = (-2, -3.5)
Hope this helps :)
+26367 Line RS passes through points R (5, 3) and S (-1, 0).
Line PQ is parallel to line RS and passes through points P (3, -1) and Q (-2, y).
What are the coordinates of point Q ?
$$\small{\text{Line PQ is parallel to line RS - cross product }}
\boxed{|(\vec{S}-\vec{R}) \times (\vec{Q}-\vec{P})| = 0}\\ \\
\left|
\left[
\binom{-1}{0}-\binom{5}{3}
\right]
\times
\left[
\binom{-2}{y}-\binom{3}{-1}
\right]
\right| = 0\\\\
\left|
\binom{-6}{-3} \times \binom{-5}{y+1}
\right| = 0\\\\
(-6)\cdot(y+1)-(-3)\cdot(-5) = 0\\
(-6)\cdot(y+1)-15 = 0\\
(-6)\cdot(y+1)=15\\\\
y+1=-\frac{15}{6}\\\\
y=-\frac{15}{6}-1\\\\
y=-\frac{21}{6}\\\\
y= -\frac{7}{2}\\\\
y=-3.5\\\\
\boxed{\vec{Q}=\binom{-2}{-3.5}}$$
