Vectors
A point only says "where". A vector says "where" and "which way, how far and how strongly": the path of a drone, the velocity of a plane, the force pushing an object. That is why in physics and in video games everything moves with vectors, not with isolated points.
Magnitude, direction and sense
A student calculated the components of the vector from point $A(2, -5)$ to point $B(-1, 4)$ by carrying out the operation $A - B$, obtaining the vector $(3, -9)$. State…
A student calculated the components of the vector from point $A(2, -5)$ to point $B(-1, 4)$ by carrying out the operation $A - B$, obtaining the vector $(3, -9)$. State the conceptual error made and calculate the correct components of vector $\vec{AB}$.
Given the points $A(-2, 5)$ and $B(4, -3)$, calculate the components of vector $\vec{AB}$ and its magnitude.
Given the points $A(-2, 5)$ and $B(4, -3)$, calculate the components of vector $\vec{AB}$ and its magnitude.
Operations with vectors
Let the vectors be $\vec{u} = (3, -1)$ and $\vec{v} = (-2, 4)$. Calculate the components of the vector resulting from the operation $3\vec{u} - 2\vec{v}$.
Let the vectors be $\vec{u} = (3, -1)$ and $\vec{v} = (-2, 4)$. Calculate the components of the vector resulting from the operation $3\vec{u} - 2\vec{v}$.
Three forces represented by vectors $\vec{F_1} = (15, -10)$, $\vec{F_2} = (-5, 25)$ and $\vec{F_3} = (-4, -3)$ in newtons act on a point object. Calculate the vector of…
Three forces represented by vectors $\vec{F_1} = (15, -10)$, $\vec{F_2} = (-5, 25)$ and $\vec{F_3} = (-4, -3)$ in newtons act on a point object. Calculate the vector of the resultant force and determine its magnitude.
Parallel vectors
Check analytically whether vectors $\vec{a} = (6, -9)$ and $\vec{b} = (-4, 6)$ are parallel (linearly dependent).
Check analytically whether vectors $\vec{a} = (6, -9)$ and $\vec{b} = (-4, 6)$ are parallel (linearly dependent).
Three telecommunication satellites are located at orbital positions $S_1(-2, 3)$, $S_2(1, 7)$ and $S_3(7, 15)$. Check analytically whether they are collinear along the…
Three telecommunication satellites are located at orbital positions $S_1(-2, 3)$, $S_2(1, 7)$ and $S_3(7, 15)$. Check analytically whether they are collinear along the same trajectory.
The scalar product
Calculate the scalar product of vectors $\vec{p} = (5, 2)$ and $\vec{q} = (-3, 4)$.
Calculate the scalar product of vectors $\vec{p} = (5, 2)$ and $\vec{q} = (-3, 4)$.
The angle between vectors and perpendicularity
Determine, using the scalar product, whether vectors $\vec{m} = (4, 6)$ and $\vec{n} = (-3, 2)$ are perpendicular.
Determine, using the scalar product, whether vectors $\vec{m} = (4, 6)$ and $\vec{n} = (-3, 2)$ are perpendicular.
Two aeroplanes depart from the same point along straight trajectories described by vectors $\vec{v_1} = (120, 160)$ and $\vec{v_2} = (200, -150)$ (in km/h). Calculate…
Two aeroplanes depart from the same point along straight trajectories described by vectors $\vec{v_1} = (120, 160)$ and $\vec{v_2} = (200, -150)$ (in km/h). Calculate the angle formed by their trajectories.
Given the triangle with vertices $A(-1, 0)$, $B(3, 2)$ and $C(1, 6)$, prove using the scalar product that it is a right-angled triangle and then calculate its area.
Given the triangle with vertices $A(-1, 0)$, $B(3, 2)$ and $C(1, 6)$, prove using the scalar product that it is a right-angled triangle and then calculate its area.