Magnitude and Direction of a Vector. Applications
A boat leaves Samaná and the captain needs to know two things: how far will it travel, and which way is the bow pointing? The magnitude tells you how far; the direction tells you which way. With these two ideas you can solve everything from the distance between two points to the effect of two forces pulling on the same post.
Magnitude of a Vector and Distance Between Points
Calculate the magnitude of the vector $\vec{v} = (3, -4)$.
Calculate the magnitude of the vector $\vec{v} = (3, -4)$.
Find the exact distance between the points $A(-2, 5)$ and $B(4, -3)$.
Find the exact distance between the points $A(-2, 5)$ and $B(4, -3)$.
Calculate the exact perimeter of the triangle whose vertices are at the coordinates $A(-1, 2)$, $B(3, 5)$ and $C(5, -1)$.
Calculate the exact perimeter of the triangle whose vertices are at the coordinates $A(-1, 2)$, $B(3, 5)$ and $C(5, -1)$.
Addition, Subtraction, and Multiplication by a Scalar
To work with forces, velocities, and displacements you need to operate with vectors. This is done component by component.
Graphically: the sum is obtained by placing the tail of $\vec v$ at the head of $\vec u$ (the sum vector goes from the tail of $\vec u$ to the head of $\vec v$), or with the parallelogram if both start at the same point. The subtraction $\vec u-\vec v$ is adding the opposite of $\vec v$. The vector joining $A$ to $B$ is $\overrightarrow{AB}=B-A$.
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…
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.
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…
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.
Direction of a Vector
The direction of a vector is that of the line containing it; the sense is which of the two ends of that line it points toward. Two vectors have the same direction if they are proportional: $\vec v = \lambda\,\vec u$.
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If $\lambda > 0$: same direction and same sense.
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If $\lambda < 0$: same direction and opposite senses.
Direction can also be measured with the slope: for $\vec v = (v_1, v_2)$, with $v_1 \neq 0$, $m = \dfrac{v_2}{v_1}$.
Determine the gradient (which indicates the direction) of the vector joining the points $P(1, -4)$ and $Q(3, 8)$.
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…
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.
Unit Vector
Obtain a unit vector that has the same direction and sense as the vector $\vec{w} = (-5, 12)$.
Obtain a unit vector that has the same direction and sense as the vector $\vec{w} = (-5, 12)$.
Find the components of a vector of magnitude $7$ that has the direction of the y-axis and a negative sense.
Find the components of a vector of magnitude $7$ that has the direction of the y-axis and a negative sense.
Going Further: Using the Unit Vector to Set a Length
The unit vector works as a "1-unit ruler" in a given direction. If you multiply that unit vector by a scalar $m$, you get a vector of length $|m|$ in that same direction (with the opposite sense if $m<0$). There is no new formula: it is the unit vector plus multiplication by a scalar.
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…
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 $\vec{u} = (-1, 1)$.
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…
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…
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.
Resultant of Forces and Velocities
When several forces or velocities act at the same time (wind on an airplane, current on a boat), the total effect is the resultant: the sum of the vectors, component by component.
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…
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.
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…
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 $\vec{T}_2$.
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…
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…
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 $k$.
Paths: Line Through Two Points and Intersection with an Axis
Some problems give a path as a line. There are two ways to describe it:
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Through two points: the vector $\overrightarrow{RS}=S-R$ is a direction vector. If $\overrightarrow{RS}=(a,b)$, the slope is $m=b/a$ and the line is $y-y_R=m(x-x_R)$.
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Through a point and a direction vector: the points are $(x_0+t\,d_1,\ y_0+t\,d_2)$, with $t$ a real number.
To find where it crosses an axis, you impose the condition of the axis: on the x-axis $y=0$; for a vertical line $x=c$ you impose $x=c$. Solve for $t$ and substitute.
Problems with Parameters and Figures
Many problems are solved by setting a magnitude (or distance) equal to a value and solving for the unknown. This leads to an equation with squares, which usually has two solutions.
Determine the coordinates of the points located on the x-axis that are at an exact distance of $5$ units from the point $P(-2, 3)$.
Determine the coordinates of the points located on the x-axis that are at an exact distance of $5$ units from the point $P(-2, 3)$.
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.
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…
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 $4$.
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…
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…
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…
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 $Q$.
Back to the beginning: the boat leaving Samaná covers a magnitude (the distance) in a direction (that of the vector); with both you know exactly where it ends up.