Operations with vectors
Two tugboats pull a ship in different directions, the wind pushes an aeroplane off course, a video game combines the movement of a character with that of a projectile: in every case you have to combine vectors. The rules for doing so are always the same, whether or not you are working on a real map.
Addition, subtraction and multiplication by a number
Given the points $A(3, -2)$ and $B(5, 4)$, calculate the coordinates of the vector $\vec{AB}$ and the vector $\vec{BA}$.
Given the points $A(3, -2)$ and $B(5, 4)$, calculate the coordinates of the vector $\vec{AB}$ and the vector $\vec{BA}$.
Given the vectors $\vec{u} = (4, -1)$ and $\vec{v} = (-2, 5)$, calculate the coordinates of the sum vector $\vec{u} + \vec{v}$ and the difference vector…
Given the vectors $\vec{u} = (4, -1)$ and $\vec{v} = (-2, 5)$, calculate the coordinates of the sum vector $\vec{u} + \vec{v}$ and the difference vector $\vec{u} - \vec{v}$.
Given the vector $\vec{w} = (3, -7)$, calculate the coordinates of the vectors $2\vec{w}$ and $-\frac{1}{2}\vec{w}$.
Given the vector $\vec{w} = (3, -7)$, calculate the coordinates of the vectors $2\vec{w}$ and $-\frac{1}{2}\vec{w}$.
Linear combination
Express vector $\vec{w} = (1, 11)$ as a linear combination of vectors $\vec{u} = (2, 3)$ and $\vec{v} = (-1, 5)$. That is, find the values of $a$ and $b$ such that…
Express vector $\vec{w} = (1, 11)$ as a linear combination of vectors $\vec{u} = (2, 3)$ and $\vec{v} = (-1, 5)$. That is, find the values of $a$ and $b$ such that $\vec{w} = a\vec{u} + b\vec{v}$.
Given points $A(1, 2)$, $B(3, -1)$ and $C(0, 4)$, find the coordinates of a point $D$ satisfying the vector equation $2\vec{AB} - 3\vec{CD} = \vec{v}$, knowing that…
Given points $A(1, 2)$, $B(3, -1)$ and $C(0, 4)$, find the coordinates of a point $D$ satisfying the vector equation $2\vec{AB} - 3\vec{CD} = \vec{v}$, knowing that vector $\vec{v}$ has coordinates $(-5, 2)$.
Unit vector
In particle physics, a proton beam follows a straight trajectory with direction vector $\vec{v} = (3, -4)$. Find a unit vector (of magnitude 1) that has the same…
In particle physics, a proton beam follows a straight trajectory with direction vector $\vec{v} = (3, -4)$. Find a unit vector (of magnitude 1) that has the same direction and sense as $\vec{v}$, and another vector that has the same direction, opposite sense and magnitude 10.
Reflection of a point
Find the coordinates of the point symmetric to $A(1, 4)$ with respect to point $M(3, -2)$.
Find the coordinates of the point symmetric to $A(1, 4)$ with respect to point $M(3, -2)$.
The vector $\vec{v} = (5, 2)$ has its initial point at $A(-3, 1)$. Determine the coordinates of its terminal point $B$.
The vector $\vec{v} = (5, 2)$ has its initial point at $A(-3, 1)$. Determine the coordinates of its terminal point $B$.
Dividing a line segment in a given ratio
Sometimes you don't want the midpoint, but a point $P$ that divides the segment $AB$ in a specific proportion, such as $\vec{AP} = 2\vec{PB}$ (the part $AP$ is twice as long as $PB$). You translate the condition into components and solve.
Calculate the coordinates of a point $P$ that divides the segment with endpoints $A(-2, 5)$ and $B(7, -1)$ such that the distance from $A$ to $P$ is twice the distance…
Calculate the coordinates of a point $P$ that divides the segment with endpoints $A(-2, 5)$ and $B(7, -1)$ such that the distance from $A$ to $P$ is twice the distance from $P$ to $B$. That is, the vector relationship $\vec{AP} = 2\vec{PB}$ holds.
Back to the beginning: If two tugboats pull with forces $\vec{F}_1 = (30, 10)\text{ kN}$ and $\vec{F}_2 = (10, 20)\text{ kN}$, the resultant force on the ship is the sum $\vec{R} = \vec{F}_1 + \vec{F}_2 = (40, 30)\text{ kN}$, with a total magnitude of $|\vec{R}| = \sqrt{40^2+30^2} = 50\text{ kN}$.