Exercises: Modulus and direction of a vector. Applications

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

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

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 $Q(3, 8)$.

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

Prove 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 with modulus $7$ that has the direction of the y-axis and a negative sense.

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 modulus of the total displacement vector of the piece from its initial position to its final position.

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

A vessel is located at point $A(2, -1)$ and moves in a straight line 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.

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 modulus of the resultant force $\vec{R} = \vec{F}_1 + \vec{F}_2$ and determine a unit vector in the direction of this resultant.

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

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

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 architect is designing a pitched roof. The profile of the roof follows the direction of the vector $\vec{u} = (4, 3)$. If due to structural restrictions the total length of the profile must be $8.5$ feet, find the components of the vector that represents this profile.

Two cables pull from 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 modulus $150$ N, which has the same direction but opposite sense to $\vec{T}_1$. Find the exact components of the vector $\vec{T}_2$.

Three coplanar forces keep an object in static equilibrium at the origin of coordinates (i.e., 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 modulus.

An airplane flies with an airspeed given by the vector $\vec{v}_a = (250, 150)$ mph. In that area, the wind blows with a velocity represented by $\vec{v}_v = (k, -50)$ mph. If we know that the actual velocity of the airplane with respect to the ground, calculated as $\vec{v}_r = \vec{v}_a + \vec{v}_v$, has an exact modulus of $300$ mph, 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 $m$ so that the rectilinear distance between $A$ and $B$ is exactly $5$ units.

An explorer robot is initially located at point $A(-3, 2)$. It is programmed to advance in a straight line in the direction and sense of the vector $\vec{u} = (2, -1)$ until it reaches 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 the vector $\vec{w} = (1, 2)$ a distance of $3\sqrt{5}$ until it stops at point $C$. Calculate the exact coordinates of the points $B$ and $C$.

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

On a coordinate plane of a video game, the main character is located at position $P(-4, 1)$. They must collect a coin located at a point $Q$ that belongs to the line passing through the 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.

Consider a parallelogram $ABCD$ with three known vertices: $A(0, 0)$, $B(6, 2)$ and $D(2, 4)$. Work out the coordinates of vertex $C$, find the modulus of its two diagonals and justify analytically whether or not it is a rhombus (i.e., whether all its sides are equal).

A light signal starts from 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 path until it hits the vertical line $x = 12$ at a point $Q$, calculate the total distance travelled by the light from $A$ to $Q$.