Projection onto a line and point-to-line distance · Short practice
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Recall
Foot of a perpendicular onto a line — objective 1
Maths4U Original · Short practice
A line is \(\mathbf r=\left[\begin{matrix}19\\2\\1\end{matrix}\right]+t\left[\begin{matrix}3\\2\\2\end{matrix}\right]\). Point P has position \(\left[\begin{matrix}24\\23\\4\end{matrix}\right]\). H is the foot of its perpendicular to the line.
The point H corresponds to parameter \(\displaystyle t=t_{0}\).
Find the foot parameter \(\displaystyle t_{0}\).
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.
Fresh review variant
Foot of a perpendicular onto a line — objective 1
Maths4U Original · Short practice
A line is \(\mathbf r=\left[\begin{matrix}20\\2\\2\end{matrix}\right]+t\left[\begin{matrix}4\\2\\3\end{matrix}\right]\). Point P has position \(\left[\begin{matrix}26\\24\\6\end{matrix}\right]\). H is the foot of its perpendicular to the line.
The point H corresponds to parameter \(\displaystyle t=t_{0}\).
Find the foot parameter \(\displaystyle t_{0}\).
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.
Method
Foot of a perpendicular onto a line — objective 2
Maths4U Original · Short practice
A line is \(\mathbf r=\left[\begin{matrix}20\\2\\2\end{matrix}\right]+t\left[\begin{matrix}4\\2\\3\end{matrix}\right]\). Point P has position \(\left[\begin{matrix}26\\24\\6\end{matrix}\right]\). H is the foot of its perpendicular to the line.
The point H corresponds to parameter \(\displaystyle t=t_{0}\).
Find the x-coordinate of H.
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.
Fresh review variant
Foot of a perpendicular onto a line — objective 2
Maths4U Original · Short practice
A line is \(\mathbf r=\left[\begin{matrix}21\\2\\3\end{matrix}\right]+t\left[\begin{matrix}1\\2\\4\end{matrix}\right]\). Point P has position \(\left[\begin{matrix}23\\25\\4\end{matrix}\right]\). H is the foot of its perpendicular to the line.
The point H corresponds to parameter \(\displaystyle t=t_{0}\).
Find the x-coordinate of H.
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.
Apply
Foot of a perpendicular onto a line — objective 3
Maths4U Original · Short practice
A line is \(\mathbf r=\left[\begin{matrix}21\\2\\3\end{matrix}\right]+t\left[\begin{matrix}1\\2\\4\end{matrix}\right]\). Point P has position \(\left[\begin{matrix}23\\25\\4\end{matrix}\right]\). H is the foot of its perpendicular to the line.
The point H corresponds to parameter \(\displaystyle t=t_{0}\).
Find the y-coordinate of H.
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.
Fresh review variant
Foot of a perpendicular onto a line — objective 3
Maths4U Original · Short practice
A line is \(\mathbf r=\left[\begin{matrix}22\\2\\1\end{matrix}\right]+t\left[\begin{matrix}2\\2\\2\end{matrix}\right]\). Point P has position \(\left[\begin{matrix}25\\26\\3\end{matrix}\right]\). H is the foot of its perpendicular to the line.
The point H corresponds to parameter \(\displaystyle t=t_{0}\).
Find the y-coordinate of H.
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.
Check understanding
Foot of a perpendicular onto a line — objective 4
Maths4U Original · Short practice
A line is \(\mathbf r=\left[\begin{matrix}22\\2\\1\end{matrix}\right]+t\left[\begin{matrix}2\\2\\2\end{matrix}\right]\). Point P has position \(\left[\begin{matrix}25\\26\\3\end{matrix}\right]\). H is the foot of its perpendicular to the line.
The point H corresponds to parameter \(\displaystyle t=t_{0}\).
Find the squared distance \(\displaystyle PH^{2}\).
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.
Fresh review variant
Foot of a perpendicular onto a line — objective 4
Maths4U Original · Short practice
A line is \(\mathbf r=\left[\begin{matrix}23\\2\\2\end{matrix}\right]+t\left[\begin{matrix}3\\2\\3\end{matrix}\right]\). Point P has position \(\left[\begin{matrix}27\\27\\5\end{matrix}\right]\). H is the foot of its perpendicular to the line.
The point H corresponds to parameter \(\displaystyle t=t_{0}\).
Find the squared distance \(\displaystyle PH^{2}\).
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.