Exercises: 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 $B(4, -3)$.

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

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 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.

To calculate the magnitude of the resultant vector of the total displacement after two consecutive stages described by $\vec{u} = (3, -4)$ and $\vec{v} = (-3, 0)$, Lucía writes on her exam:

1. a) Identify and explain the two errors made by Lucía: one algebraic-geometric concerning the magnitude of the sum of vectors, and another conceptual concerning the difference between the total distance traveled along the path and the magnitude of the resulting displacement vector.
2. b) Correctly calculate the components of the resultant vector $\vec{u} + \vec{v}$ and its exact magnitude. Geometrically interpret why this result is strictly less than the value obtained by Lucía.

Consider the vector $\vec{w} = (-8, 15)$.

1. a) Without using a calculator or solving the square root, estimate with reasoning between which two integers the value of $|\vec{w}|$ must lie. Rely for this on two geometric properties of the right triangle formed by its components: the hypotenuse is always greater than either leg, but less than the sum of both. Furthermore, reason whether you expect the magnitude to be closer to the lower bound or the upper bound.
2. b) Calculate the exact value of $|\vec{w}|$ by applying the magnitude formula and check whether it confirms the estimate you made.

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.

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.

Calculate the exact perimeter of the triangle whose vertices are at the 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 $\vec{v} = (3, 4)$ 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.

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 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 $8,5$ metres, 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 $\vec{T}_2$.

A Maritime Rescue vessel based in Tarifa (Cádiz) departs towards an intervention zone following the direction and sense of the vector $\vec{d} = (12, 5)$ (coordinates measured in nautical miles). Due to protocol and fuel considerations, it is ordered to advance in that exact direction until completing a distance of $7,8$ nautical miles.

1. a) Model the patrol boat's displacement by finding the exact components of a vector $\vec{r}$ that has a magnitude of $7,8$ nautical miles and the same direction and sense as $\vec{d}$.
2. b) During the journey, the currents of the Strait cause a drift described by the vector $\vec{c} = (-1,2;\, 1,0)$ nautical miles. Calculate the actual final position vector relative to the Tarifa base and determine at what straight-line distance the vessel is from its starting point (round the result to two decimal places).

Given the point $A(2, -1)$, the goal is to determine the coordinates of a point $B$ belonging to the x-axis (that is, whose y-coordinate is zero) such that the distance between $A$ and $B$ is exactly $\sqrt{26}$ units.

1. a) Set up the equation for the magnitude of the vector $\vec{AB}$ in terms of the unknown x-coordinate of $B$ and find all possible solutions for that point.
2. b) For each of the positions of $B$ found, calculate the unit vector that has the same direction and sense as $\vec{AB}$, and explain the geometric symmetry relationship that exists between the obtained unit vectors.

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.

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.

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 $C$.

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.

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).

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 $Q$.

We want to determine the coordinates of a point $P$ located at a distance of $15$ units from the point $A(-1, 3)$, such that the vector $\vec{AP}$ has the same direction and sense as the vector $\vec{v} = (-6, 8)$. Two students in the class propose different approaches:

• Marcos's method: He writes the parametric coordinates of the points on the line passing through $A$ with direction vector $\vec{v}$, $P(-1 - 6t,\, 3 + 8t)$, sets up the Euclidean distance equation $d(A, P) = 15$, and solves for the parameter $t$.

• Sara's method: She finds the unit vector $\vec{u}$ in the direction and sense of $\vec{v}$, scales this vector by the desired distance calculating $\vec{w} = 15\cdot\vec{u}$, and finds $P$ by adding $\vec{w}$ to the position of $A$.

1. a) Apply both methods step by step and verify that exactly the same point $P$ is reached.
2. b) Critically compare both procedures: justify which one is more direct and has a lower algebraic cost. Likewise, explain what specific modification should be introduced into each of the two methods if the point were required to be located at the same distance but in the opposite sense.

In a civil engineering structural test, a ball joint located at the origin supports the coplanar action of two forces, $\vec{F}_1$ and $\vec{F}_2$, whose known magnitudes are $|\vec{F}_1| = 6\text{ N}$ and $|\vec{F}_2| = 8\text{ N}$.

1. a) Formally prove, using the geometric properties of vectors, that the magnitude of the resultant force $\vec{R} = \vec{F}_1 + \vec{F}_2$ cannot take any arbitrary value, but must necessarily satisfy the inequality $2\text{ N} \le |\vec{R}| \le 14\text{ N}$. Describe precisely what relative orientation $\vec{F}_1$ and $\vec{F}_2$ must have for the magnitude to reach each of these extreme values.
2. b) If the force $\vec{F}_1$ acts horizontally in the positive direction, that is, $\vec{F}_1 = (6, 0)\text{ N}$, and the load sensor records that the magnitude of the total resultant is exactly $|\vec{R}| = 10\text{ N}$, set up a system with the equations for the magnitudes of $\vec{F}_2$ and $\vec{R}$ to algebraically calculate the components of all possible forces $\vec{F}_2$ compatible with the measurement. Obtain, for each case, its corresponding unit vector.