Vectors: Definitions and Basic Operations
A kayaker aims straight across a river at $4$ mph, but the current pushes her downstream at $3$ mph. Where does she end up, and how fast is she really moving? A single number can't answer that. You also need to know which way each speed points. The tool that keeps track of both is the vector.
Scalars and vectors
Speed and velocity are a good pair to remember. Speed ($60$ mph) is a scalar. Velocity ($60$ mph heading east) is a vector. The speed is the magnitude of the velocity.
Classify the following quantities used in physics and geometry as scalars or vectors, briefly justifying your answer based on their definition: Temperature of a data…
Classify the following quantities used in physics and geometry as scalars or vectors, briefly justifying your answer based on their definition:
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Temperature of a data server ($45^\circ \text{C}$).
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Wind velocity in a wind farm ($20 \text{ km/h}$ in a Northwest direction).
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Mass of an artificial satellite ($1200 \text{ kg}$).
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Displacement of a drone from its base to a delivery point.
Component form of a vector
To compute with vectors, we place them in the coordinate plane. A vector is drawn as an arrow, and its initial point is the tail while its terminal point is the tip.
Many problems write the components with ordinary parentheses, $(a, b)$, instead of $\langle a, b\rangle$. It means the same vector.
So $\vec{AB} = \langle 5-1,\ 4-1\rangle = \langle 4, 3\rangle$.
A robotic arm on an assembly line moves in a two-dimensional Cartesian plane. If its initial position is the point $A(3, -2)$ and its final position after a movement is…
A robotic arm on an assembly line moves in a two-dimensional Cartesian plane. If its initial position is the point $A(3, -2)$ and its final position after a movement is $B(-1, 5)$, determine the components of the displacement vector $\vec{AB}$.
A fishing boat is located at coordinates $P(-4, 2)$ on a nautical chart. If the captain programs the autopilot to make a rectilinear displacement defined by the vector…
A fishing boat is located at coordinates $P(-4, 2)$ on a nautical chart. If the captain programs the autopilot to make a rectilinear displacement defined by the vector $\vec{v} = (8, -3)$, determine the coordinates of the destination point $Q$.
Magnitude of a vector
Drag the point below and watch the components form a right triangle.
In the design of a video game, the velocity of a character is given by the vector $\vec{v} = (-6, 8)$ in pixels per frame. Calculate the modulus of this vector to…
In the design of a video game, the velocity of a character is given by the vector $\vec{v} = (-6, 8)$ in pixels per frame. Calculate the modulus of this vector to determine the actual speed of the character on the screen.
The force applied to an object is given by the vector $\vec{F} = (5, k)$ in Newtons. If a sensor indicates that the total magnitude of the force is $13 \text{ N}$,…
The force applied to an object is given by the vector $\vec{F} = (5, k)$ in Newtons. If a sensor indicates that the total magnitude of the force is $13 \text{ N}$, determine the possible values of the parameter $k$.
Distance between two points
The distance from $A$ to $B$ is the length of the vector $\vec{AB}$:
$$AB = \|\vec{AB}\| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$This gives a way to find points that are a given distance from a known point. Points on the $x$-axis look like $(x, 0)$.
To install a fiber optic cable between two buildings, surveyors mark the positions of the connections on a plan as $A(-3, 5)$ and $B(2, -7)$, with units in feet.…
To install a fiber optic cable between two buildings, surveyors mark the positions of the connections on a plan as $A(-3, 5)$ and $B(2, -7)$, with units in feet. Calculate the straight-line distance between both points.
Find the coordinates of all the points located on the x-axis ($X$-axis) whose distance to the point $A(-2, 3)$ is exactly equal to $5$ units.
Find the coordinates of all the points located on the x-axis ($X$-axis) whose distance to the point $A(-2, 3)$ is exactly equal to $5$ units.
Equal vectors
Some problems call equal vectors equipollent, use modulus for magnitude, and sense for which way along a line a vector points. In this course we say equal vectors, magnitude, and direction.
A useful consequence: $ABCD$ is a parallelogram exactly when $\vec{AB} = \vec{DC}$ (opposite sides are equal vectors).
Observe the vectors represented on the following grid. Identify which pairs of vectors are equipollent (equal), which have the same modulus but different direction, and…
Observe the vectors represented on the following grid. Identify which pairs of vectors are equipollent (equal), which have the same modulus but different direction, and which have the same direction but opposite sense.
Three lighting drones for a night show are located at coordinates $A(1, 2)$, $B(5, 3)$ and $C(6, 7)$. For the geometric figure they form in the sky to be a parallelogram…
Three lighting drones for a night show are located at coordinates $A(1, 2)$, $B(5, 3)$ and $C(6, 7)$. For the geometric figure they form in the sky to be a parallelogram $ABCD$, determine the coordinates of the position where the fourth drone $D$ must be located.
Parallel vectors
Multiplying a vector by a number $k$ (a scalar) multiplies each component by $k$: $k\langle a, b\rangle = \langle ka, kb\rangle$. The result is $|k|$ times as long.
To test whether $\vec u = \langle a_1, a_2\rangle$ and $\vec v = \langle b_1, b_2\rangle$ are parallel, check that the components are proportional by cross-multiplying:
$$a_1 b_2 = a_2 b_1$$(This works even if a component is $0$, where a division would fail.) Then the sign of $k$ tells you whether they point the same way or opposite ways.
Collinear points. Three points $A$, $B$, $C$ lie on one line exactly when $\vec{AB}$ and $\vec{AC}$ are parallel.
Two model airplanes follow trajectories defined by the vectors $\vec{u} = (4, -6)$ and $\vec{v} = (-2, 3)$. Prove, using the definition of parallel vectors, whether both…
Two model airplanes follow trajectories defined by the vectors $\vec{u} = (4, -6)$ and $\vec{v} = (-2, 3)$. Prove, using the definition of parallel vectors, whether both airplanes fly in the same direction. Do they have the same sense?
To lay out a straight road, engineers must ensure that three control points are perfectly aligned. The first two points are at $A(-1, 3)$ and $B(2, 5)$. If the third…
To lay out a straight road, engineers must ensure that three control points are perfectly aligned. The first two points are at $A(-1, 3)$ and $B(2, 5)$. If the third point has coordinate $y = 9$, find its coordinate $x$ using the concept of parallel vectors.
Determine the values of the parameters $a$ and $b$ knowing that the vectors $\vec{u} = (a, 3)$ and $\vec{v} = (-2, b)$ have the same direction (they are parallel),…
Determine the values of the parameters $a$ and $b$ knowing that the vectors $\vec{u} = (a, 3)$ and $\vec{v} = (-2, b)$ have the same direction (they are parallel), opposite sense, and furthermore it is satisfied that the modulus of $\vec{u}$ is $5$.
Adding vectors and position vectors
Why? Two displacements in a row move you right by $a + c$ and up by $b + d$, so the total is just the sum of components.
Now we can close the kayak story. Take east as $+x$ and north (straight across) as $+y$.
Position vectors
Since $\vec{OA} + \vec{AB} = \vec{OB}$, we get a very useful formula:
$$\vec{AB} = \vec{OB} - \vec{OA}$$A Martian exploration robot makes two consecutive displacements. First, it moves from its base $O(0,0)$ to point $P$ defined by the vector $\vec{v_1} = (12, 5)$. Then,…
A Martian exploration robot makes two consecutive displacements. First, it moves from its base $O(0,0)$ to point $P$ defined by the vector $\vec{v_1} = (12, 5)$. Then, from $P$, it makes a second displacement $\vec{v_2} = (-4, 10)$ until it reaches point $Q$. Calculate the coordinates of point $Q$ and the total straight-line distance from the base $O$ to $Q$.
Given the regular hexagon $ABCDEF$ in the figure, whose center is the origin of coordinates $O(0,0)$ and vertex $A$ has coordinates $(2,0)$. Express the vectors…
Given the regular hexagon $ABCDEF$ in the figure, whose center is the origin of coordinates $O(0,0)$ and vertex $A$ has coordinates $(2,0)$. Express the vectors $\vec{AC}$ and $\vec{AD}$ as a linear combination of the position vectors of the vertices (for example, by operating with points or adding known vectors from the figure).
A boat attempts to cross a river perpendicularly to the banks with its own velocity given by the vector $\vec{v_b} = (0, 12)$ mph. However, the river current exerts a…
A boat attempts to cross a river perpendicularly to the banks with its own velocity given by the vector $\vec{v_b} = (0, 12)$ mph. However, the river current exerts a constant velocity represented by the vector $\vec{v_c} = (5, 0)$ mph. Calculate the actual velocity vector of the boat with respect to an observer on the bank and determine its actual speed (modulus of the resultant vector).
Midpoints and dividing a segment
The same idea works for any fraction of the way: a point a fraction $t$ of the way from $A$ to $B$ is $A + t\,\vec{AB}$.
Some problems give a vector relationship instead of a ratio, such as $\vec{PA} + 3\vec{PB} = \vec 0$. Write each vector as (end) minus (start) and solve for $P$.
A support pillar is to be placed exactly in the center of a steel beam. If the ends of the beam are located at coordinates $M(-1, 4)$ and $N(7, -2)$, determine the…
A support pillar is to be placed exactly in the center of a steel beam. If the ends of the beam are located at coordinates $M(-1, 4)$ and $N(7, -2)$, determine the coordinates of the point where the pillar must go.
An architect designs a straight staircase that joins point $P(-2, 1)$ on the ground with point $Q(7, -5)$ on a platform. If they wish to place two intermediate supports…
An architect designs a straight staircase that joins point $P(-2, 1)$ on the ground with point $Q(7, -5)$ on a platform. If they wish to place two intermediate supports that divide the staircase into three sections of equal length, calculate the coordinates where these supports must be anchored.
Given the points $A(1, -2)$ and $B(6, 3)$, find the coordinates of a point $P$ located on the line segment joining them, such that the vector relationship…
Given the points $A(1, -2)$ and $B(6, 3)$, find the coordinates of a point $P$ located on the line segment joining them, such that the vector relationship $2\vec{PA} + 3\vec{PB} = \vec{0}$ is satisfied.
Unit vectors
Unit vectors are handy for "direction only" questions. A vector of magnitude $m$ in the direction of $\vec v$ is $m\hat v$. A vector of magnitude $m$ in the opposite direction is $-m\hat v$.
When the magnitude is irrational, rationalize the denominator. For $\vec w = \langle 1, 1\rangle$: $\hat w = \left\langle \tfrac{1}{\sqrt2}, \tfrac{1}{\sqrt2}\right\rangle = \left\langle \tfrac{\sqrt2}{2}, \tfrac{\sqrt2}{2}\right\rangle$.
Angle bisectors
Two unit vectors have the same length, so they form a rhombus. The diagonal of a rhombus bisects its angle. Therefore $\hat u + \hat v$ points along the bisector of the angle between $\vec u$ and $\vec v$.
In a physics simulator, the wind direction is given by the vector $\vec{w} = (-3, 4)$. To correctly apply aerodynamic formulas, the game engine needs a unit vector that…
In a physics simulator, the wind direction is given by the vector $\vec{w} = (-3, 4)$. To correctly apply aerodynamic formulas, the game engine needs a unit vector that maintains exactly the same direction and sense as $\vec{w}$. Calculate it.
Find a vector $\vec{v}$ that is parallel to the vector $\vec{u} = (1, -2)$, has the opposite sense to $\vec{u}$, and whose modulus is exactly $10$.
Find a vector $\vec{v}$ that is parallel to the vector $\vec{u} = (1, -2)$, has the opposite sense to $\vec{u}$, and whose modulus is exactly $10$.
Given the vectors $\vec{u} = (4, 0)$ and $\vec{v} = (0, 3)$. We wish to find a vector $\vec{w}$ that is the angle bisector of the angle formed by $\vec{u}$ and…
Given the vectors $\vec{u} = (4, 0)$ and $\vec{v} = (0, 3)$. We wish to find a vector $\vec{w}$ that is the angle bisector of the angle formed by $\vec{u}$ and $\vec{v}$. Knowing that the sum of two unit vectors results in a vector in the direction of the angle bisector, calculate the exact coordinates of a direction vector $\vec{w}$ of this angle bisector.
Using vectors in geometry
Vectors turn geometric claims into short calculations.
Symmetric point. The reflection $A'$ of $A$ through a point $P$ is the point for which $P$ is the midpoint of $AA'$. That means $\vec{AP} = \vec{PA'}$, so $A' = 2P - A$.
Proofs with vectors. Put the position vector $\vec a$ for point $A$, $\vec b$ for $B$, and so on. Midpoints become averages, and equal vectors prove parallel, equal-length sides.
Find the coordinates of the symmetric point $A'$ of point $A(3, -1)$ with respect to point $P(1, 2)$. Set up the solution using the definition of a vector joining two…
Find the coordinates of the symmetric point $A'$ of point $A(3, -1)$ with respect to point $P(1, 2)$. Set up the solution using the definition of a vector joining two points and indicate the geometric relationship between the vectors $\vec{AP}$ and $\vec{PA'}$.
The points $A(1, 1)$, $B(4, 2)$, $C(3, 5)$ and $D(0, 4)$ are given. Calculate the vectors that form the sides of the quadrilateral $ABCD$ and the lengths of its…
The points $A(1, 1)$, $B(4, 2)$, $C(3, 5)$ and $D(0, 4)$ are given. Calculate the vectors that form the sides of the quadrilateral $ABCD$ and the lengths of its diagonals. Based on these calculations (and without using the scalar product), justify whether the figure is a rhombus, a square, a rectangle or none of these regular figures.
Given any quadrilateral with vertices $A, B, C$ and $D$. Let $M, N, P$ and $Q$ be the midpoints of sides $AB, BC, CD$ and $DA$ respectively. Prove, using exclusively…
Given any quadrilateral with vertices $A, B, C$ and $D$. Let $M, N, P$ and $Q$ be the midpoints of sides $AB, BC, CD$ and $DA$ respectively. Prove, using exclusively operations with vectors and coordinates of points, that the quadrilateral $MNPQ$ is always a parallelogram.
Summary
Back to the kayaker: her velocity across is $\langle 0, 4\rangle$ and the current adds $\langle 3, 0\rangle$, so she actually moves with velocity $\langle 3, 4\rangle$, at a speed of $5$ mph. She drifts $3$ miles downstream for every $4$ miles she travels across the river.