Magnitude and Direction of a Vector. Applications

Calculate the magnitude of the vector $\vec{v} = (3, -4)$.

Find the exact distance between the points $A(-2, 5)$ and $B(4, -3)$.

A chess player moves their pawn from the square represented by point $(1, 2)$ to square $(1, 4)$, and then moves a knight from $(1, 4)$ to $(3, 5)$. Calculate the magnitude of the total displacement vector of the piece from its initial to its final position.

Calculate the exact perimeter of the triangle whose vertices are at the coordinates $A(-1, 2)$, $B(3, 5)$ and $C(5, -1)$.

Obtain a unit vector that has the same direction and sense as the vector $\vec{w} = (-5, 12)$.

Determine the gradient (which indicates the direction) of the vector joining the points $P(1, -4)$ and $Q(3, 8)$.

Check analytically whether the vectors $\vec{u} = (2, -5)$ and $\vec{v} = (-4, 10)$ have the same direction. If so, indicate whether they have the same sense or opposite senses.

Find the components of a vector of magnitude $7$ that has the direction of the y-axis and a negative sense.

A drone travels in a straight line according to the vector $\vec{d} = (k, 6)$. If the total distance traveled is $10$ meters, calculate the possible values of the parameter $k$.

Find the coordinates of a vector $\vec{v}$ knowing that its magnitude is $5\sqrt{2}$ and that it has the same direction and sense as the vector $\vec{u} = (-1, 1)$.

A boat is at the point $A(2, -1)$ and travels in a straight line for a distance of $13$ nautical miles. If the direction vector of its trajectory is proportional to $\vec{d} = (5, 12)$ and it moves in the same sense as $\vec{d}$, determine the exact coordinates of the arrival point.

An architect is designing a sloping roof. The profile of the roof follows the direction of the vector $\vec{u} = (4, 3)$. If due to structural constraints the total length of the profile must be $8.5$ meters, find the components of the vector representing this profile.

Two cables pull on the top of a pole. The first cable exerts a force represented by the vector $\vec{T}_1 = (120, 160)$ N. The second cable exerts a force $\vec{T}_2$ of magnitude $150$ N, which has the same direction but opposite sense to $\vec{T}_1$. Find the exact components of the vector $\vec{T}_2$.

An exploration robot is initially at point $A(-3, 2)$. It is programmed to move in a straight line in the direction and sense of vector $\vec{u} = (2, -1)$ until reaching a point $B$, such that the distance $AB$ is equal to $4\sqrt{5}$. Next, it turns and advances in the direction and sense of vector $\vec{w} = (1, 2)$ for a distance of $3\sqrt{5}$ until stopping at point $C$. Calculate the exact coordinates of points $B$ and $C$.

Two coplanar forces represented by the vectors $\vec{F}_1 = (30, -40)$ N and $\vec{F}_2 = (-15, 20)$ N act on a body. Calculate the magnitude of the resultant force $\vec{R} = \vec{F}_1 + \vec{F}_2$ and determine a unit vector in the direction of this resultant.

In a physics simulation, a particle starts from the origin of coordinates and moves with a constant velocity vector $\vec{v} = (3, 4)$ m/s for $2$ seconds. Immediately afterwards, it changes its velocity to $\vec{w} = (-1, 2)$ m/s and maintains this motion for a further $3$ seconds. Calculate how far from the origin the particle is at the end of its journey.

Three coplanar forces maintain an object in static equilibrium at the origin of coordinates (that is, the resultant force is the zero vector). The components of two of them are known: $\vec{F}_1 = (40, 30)$ N and $\vec{F}_2 = (-20, 50)$ N. Calculate the components of the third force $\vec{F}_3$ and its exact magnitude.

Determine the coordinates of the points located on the x-axis that are at an exact distance of $5$ units from the point $P(-2, 3)$.

An aircraft flies with a velocity relative to the air given by the vector $\vec{v}_a = (250, 150)$ km/h. In that area, the wind blows with a velocity represented by $\vec{v}_v = (k, -50)$ km/h. If we know that the actual velocity of the aircraft relative to the ground, calculated as $\vec{v}_r = \vec{v}_a + \vec{v}_v$, has an exact magnitude of $300$ km/h, determine the possible values of the parameter $k$.

Given the points $A(1, 2)$ and $B(m, m-1)$, find the real value or values of $m$ such that the straight-line distance between $A$ and $B$ is exactly $5$ units.

Find analytically all vectors $\vec{v} = (x, y)$ of magnitude $\sqrt{10}$ that satisfy the condition that the sum of their two components is exactly $4$.

On a video game coordinate plane, the main character is at position $P(-4, 1)$. They must collect a coin located at a point $Q$ belonging to the line passing through points $R(0, 5)$ and $S(2, 7)$. If the character moves exactly $3\sqrt{2}$ units in a straight line to reach $Q$, find the possible coordinates where the coin is located.

A light signal leaves point $A(0, 4)$ with a direction vector $\vec{d} = (2, -1)$. Upon reaching the x-axis at a point $P$, it reflects, changing its direction and sense according to the new direction vector $\vec{d}' = (2, 1)$. If the signal continues its trajectory until hitting the vertical line $x = 12$ at a point $Q$, calculate the total distance travelled by the light from $A$ to $Q$.

Consider a rhomboid $ABCD$ for which three of its vertices are known: $A(0, 0)$, $B(6, 2)$ and $D(2, 4)$. Calculate the coordinates of vertex $C$, obtain the magnitude of its two diagonals and justify analytically whether or not it is a rhombus (that is, whether all its sides are equal).