IB Mathematics AHL 3.10 Concept of a vector AI HL Paper 1- Exam Style Questions- New Syllabus
▶️ Answer/Explanation
(a)
The given direction vector of the second airplane is
\(\begin{pmatrix}7\\-24\\0\end{pmatrix}\).
Its magnitude is
\(\sqrt{7^2+(-24)^2+0^2}=\sqrt{49+576}=25\).
Therefore, the unit vector in this direction is
\(\dfrac{1}{25}\begin{pmatrix}7\\-24\\0\end{pmatrix}\).
Since the airplane travels at \(800\text{ km h}^{-1}\), its velocity vector is
\(\dfrac{800}{25}\begin{pmatrix}7\\-24\\0\end{pmatrix}=32\begin{pmatrix}7\\-24\\0\end{pmatrix}=\begin{pmatrix}224\\-768\\0\end{pmatrix}\).
Using \(\text{position}=\text{initial position}+t(\text{velocity})\),
\(r_2=\begin{pmatrix}-13\\50\\7.2\end{pmatrix}+t\begin{pmatrix}224\\-768\\0\end{pmatrix}\).
Equivalently,
\(r_2=\begin{pmatrix}-13\\50\\7.2\end{pmatrix}+32t\begin{pmatrix}7\\-24\\0\end{pmatrix}\).
✅ Answer: \(r_2=\begin{pmatrix}-13\\50\\7.2\end{pmatrix}+t\begin{pmatrix}224\\-768\\0\end{pmatrix}\)
(b)
The relative position vector from the second airplane to the first airplane is
\(r_1-r_2\).
\(r_1-r_2=\begin{pmatrix}29\\-32\\5.8\end{pmatrix}-\begin{pmatrix}-13\\50\\7.2\end{pmatrix}+t\left[\begin{pmatrix}-320\\600\\0.5\end{pmatrix}-\begin{pmatrix}224\\-768\\0\end{pmatrix}\right]\).
Therefore,
\(r_1-r_2=\begin{pmatrix}42\\-82\\-1.4\end{pmatrix}+t\begin{pmatrix}-544\\1368\\0.5\end{pmatrix}\).
The distance between the airplanes is therefore
\(d(t)=\sqrt{(42-544t)^2+(-82+1368t)^2+(-1.4+0.5t)^2}\).
It is easier to minimize the square of the distance:
\(D(t)=(42-544t)^2+(-82+1368t)^2+(-1.4+0.5t)^2\).
Differentiating and setting the derivative equal to zero gives
\(2(42-544t)(-544)+2(-82+1368t)(1368)+2(-1.4+0.5t)(0.5)=0\).
Solving gives
\(t=0.0622991\ldots\text{ hours}\).
This is approximately \(3.74\) minutes after monitoring begins.
Substituting this value into the distance expression:
\(d_{\min}=\sqrt{(42-544(0.0622991\ldots))^2+(-82+1368(0.0622991\ldots))^2+(-1.4+0.5(0.0622991\ldots))^2}\).
\(d_{\min}=8.83380\ldots\text{ km}\).
✅ Answer: The least distance between the two airplanes is \(8.83\text{ km}\).