Level 1 – Understanding and direct application (minimum standard for the Baccalaureate)
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In the coordinate system $xOy$, the points $A(2,3)$ and $B(4,7)$ are given. Find the coordinates of the vector $\overrightarrow{AB}$ and its magnitude.
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The vectors $\overrightarrow{u} = 3\overrightarrow{i} - 2\overrightarrow{j}$ and $\overrightarrow{v} = - \overrightarrow{i} + 4\overrightarrow{j}$ are given. Calculate the coordinates of the vector $\overrightarrow{w} = 2\overrightarrow{u} + 3\overrightarrow{v}$.
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Find the length of segment $AB$ where $A( - 1,4)$ and $B(5, - 4)$.
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Find the coordinates of the midpoint of segment $AB$, where $A( - 3,5)$ and $B(7,1)$.
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Find the real number $a$ for which the vectors $\overrightarrow{u} = 2\overrightarrow{i} + 4\overrightarrow{j}$ and $\overrightarrow{v} = a\overrightarrow{i} + 6\overrightarrow{j}$ are collinear.
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Consider the vectors $\overrightarrow{u} = 4\overrightarrow{i} + 3\overrightarrow{j}$ and $\overrightarrow{v} = a\overrightarrow{i} - 8\overrightarrow{j}$. Find $a \in \mathbb{R}$ such that $\overrightarrow{u}\bot\overrightarrow{v}$.
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Find the slope of the line passing through the points $A(1,2)$ and $B(4,8)$.
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Write the equation of the line passing through the point $A(2, - 1)$ with slope $m = - 3$.
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Find the equation of the line passing through the points $A(2,3)$ and $B( - 1,0)$.
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Write the explicit equation of the line $d:2x - 3y + 6 = 0$.
Identify its slope.
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The line $d:y = 2x - 5$ is given. Find the equation of the line $d'$ passing through $A(1,4)$ and parallel to $d$.
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The line $d:y = - \frac{1}{3}x + 2$ is given. Find the slope of a line perpendicular to $d$.
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Calculate the distance from the point $A(3,2)$ to the line $d:3x + 4y - 2 = 0$.
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Find the centroid of the triangle with vertices $A(1,2)$, $B(5,0)$ and $C(3,7)$.
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Check whether the points $A(1,1)$, $B(2,3)$ and $C(4,7)$ are collinear.
Level 2 – Average application (standard format, item I5 of the BAC)
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Find the equation of the line passing through the point $A(3, - 2)$ and parallel to the line $x + 2y - 5 = 0$.
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Find the equation of the line passing through $P( - 1,2)$ and perpendicular to the line $d:2x - y + 3 = 0$.
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In the Cartesian coordinate system $xOy$, consider the points $A(1,2)$, $B(3,4)$ and $C(5,0)$. Find the length of the median from $A$ of triangle $ABC$.
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Consider the points $A(2,1)$, $B(6,3)$ and $C(4,5)$. Write the equation of the altitude from vertex $C$ in triangle $ABC$.
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Find the real number $m$ for which the lines $d_{1}:mx + y - 2 = 0$ and $d_{2}:4x + my + 1 = 0$ are parallel.
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Find the real number $a$ for which the lines $d_{1}:(a + 1)x + 3y - 1 = 0$ and $d_{2}:2x - y + 4 = 0$ are perpendicular.
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In the coordinate system $xOy$, the points $A(1,3)$, $B(5,1)$ and $C(3,7)$ are given. Calculate the area of triangle $ABC$.
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Find the coordinates of the point $D$ such that the quadrilateral $ABCD$ is a parallelogram, knowing that $A(1,1)$, $B(4,2)$ and $C(5,5)$.
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Let the vectors $\overrightarrow{u} = 2\overrightarrow{i} + (m - 1)\overrightarrow{j}$ and $\overrightarrow{v} = (m + 1)\overrightarrow{i} + 4\overrightarrow{j}$. Find $m > 0$ such that $|\overrightarrow{u}| = |\overrightarrow{v}|$.
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The points $A( - 2,1)$ and $B(4,5)$ are given. Find the equation of the perpendicular bisector of segment $\lbrack AB\rbrack$.
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Find the coordinates of the reflection of the point $A(3,4)$ with respect to the point $B(1,1)$.
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Find the slope of the line $(\alpha + 1)x + (\alpha - 1)y + 3 = 0$, where $\alpha \in \mathbb{R} \smallsetminus \left\{ 1 \right\}$.
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Show that the triangle with vertices $A(0,0)$, $B(3,1)$ and $C( - 1,3)$ is right and isosceles.
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The vector $\overrightarrow{u} = 5\overrightarrow{i} - 12\overrightarrow{j}$ is given. Find a vector of length 1 (a unit vector) with the same direction and sense as $\overrightarrow{u}$.
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Find the angle formed by the vectors $\overrightarrow{u} = \overrightarrow{i} + \sqrt{3}\overrightarrow{j}$ and $\overrightarrow{v} = \sqrt{3}\overrightarrow{i} + \overrightarrow{j}$.
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Find $a \in \mathbb{R}$ such that the point $P(1,a)$ is at distance $2$ from the line $d:3x - 4y + 1 = 0$.
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The points $A(2,3)$, $B(4,y)$ and $C(x,7)$ are such that $B$ is the midpoint of $\lbrack AC\rbrack$. Find $x$ and $y$.
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Calculate the dot product $\overrightarrow{u} \cdot \overrightarrow{v}$, knowing that $||\overrightarrow{u}|| = 4$, $||\overrightarrow{v}|| = 3$ and the angle between them is $60^{\circ}$.
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Find the equations of the lines parallel to $d:4x - 3y + 1 = 0$ at distance $2$ from it.
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In triangle $ABC$, the point $A(1,1)$ and the equations of the perpendicular bisectors $x - y + 1 = 0$ and $2x + y - 4 = 0$ are known. Find the circumcenter of the triangle.
Level 3 – Deepening and consolidation (integrated mathematical reasoning)
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Consider the lines $d_{m}:(2m + 1)x + (m - 1)y + 3m - 2 = 0$, $m \in \mathbb{R}$. Show that all the lines $d_{m}$ pass through a fixed point $P$ and find its coordinates.
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In $\bigtriangleup ABC$, the vertices $A(2,4)$, $B( - 1,1)$ and $C(5, - 1)$ are given. Find the coordinates of the orthocenter $H$ of the triangle.
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Find the coordinates of the reflection of the point $A(1,5)$ across the line $d:2x - y + 1 = 0$.
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Let $A(1,2)$ and $B(3,0)$. Find the coordinates of the point $C$ on the first angle bisector ($y = x$) such that the area of triangle $ABC$ equals $4$.
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Let the points $A(2,0)$ and $B(0,4)$. Find the coordinates of the points $M$ on the $Ox$ axis for which the angle $\widehat{AMB}$ is a right angle ($90^{\circ}$).
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Find $a \in \mathbb{R}$ such that the vectors $\overrightarrow{u} = (a + 1)\overrightarrow{i} + 2\overrightarrow{j}$ and $\overrightarrow{v} = 3\overrightarrow{i} + (a - 1)\overrightarrow{j}$ form an obtuse angle.
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In triangle $ABC$, the point $M$ divides side $BC$ in the ratio $\frac{BM}{MC} = 2$. Express the vector $\overrightarrow{AM}$ in terms of the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$.
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Let $ABCD$ be any quadrilateral. Show using vectors that $\overrightarrow{AB} + \overrightarrow{CD} = \overrightarrow{AD} + \overrightarrow{CB}$.
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Find the coordinates of the orthogonal projection of the origin $O(0,0)$ onto the line $d:3x - 4y + 25 = 0$.
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Consider the lines $d_{1}:y = 2x + 1$, $d_{2}:y = - x + 4$ and $d_{3}:y = mx + 3$. Find $m \in \mathbb{R}$ such that the three lines are concurrent.
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Consider the points $A(a,0)$, $B(0,b)$ with $a,b > 0$ such that the area of triangle $OAB$ is $6$. Knowing that the line $AB$ has slope $m = - \frac{3}{2}$, find $a$ and $b$.
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Show that a point $P(x,y)$ equally distant from the lines $d_{1}:3x - 4y + 1 = 0$ and $d_{2}:5x + 12y - 2 = 0$ lies on one of the angle bisectors of the angles formed by the two lines, and write the equations of these bisectors.
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Consider the position vectors of the vertices of $\bigtriangleup ABC$: $\overrightarrow{r_{A}} = \overrightarrow{i} + 2\overrightarrow{j}$, $\overrightarrow{r_{B}} = 4\overrightarrow{i} + 6\overrightarrow{j}$, $\overrightarrow{r_{C}} = 7\overrightarrow{i} - 1\overrightarrow{j}$. Find the magnitude of the vector $\overrightarrow{v} = \overrightarrow{r_{A}} + \overrightarrow{r_{B}} - 2\overrightarrow{r_{C}}$.
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Find the equation of the line passing through the intersection point of the lines $x - y + 1 = 0$ and $2x + y - 4 = 0$ and parallel to the first angle bisector.
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In $\bigtriangleup ABC$, the point $A(1,3)$ and the equations of the perpendicular bisectors of the sides $AB$ and $AC$: $x + y - 3 = 0$ and $x - 2y + 1 = 0$, respectively, are given. Find the coordinates of the vertices $B$ and $C$.